TRIGONOMETRIC FUNCTIONS CHAPTER 3
An angle is generated by revolving a ray from the initial position OX to a terminal side OP. Here, the line OX is called initial side and OP is called terminal side of the angle.
An angle is positive, if the direction of rotation is anti-clockwise. An angle is negative, if the direction of rotation is clockwise.
1. Measurement of angles: There are three systems of measurement of angles.
i. Sexagesimal measure
ii. Centesimal measure
iii. Radian measure
2. Sexagesimal measure or degree measure: An angle is said to have a measure of one degree, if a rotation from the initial side to terminal side is th 1 360
It is represented as 1°.
1° = 60 ' (60 minutes),
1' = 60" (60 seconds)
of a revolution.
3. Centesimal Measure: In this system, a right angle is subdivided into 100 grades, with each grade further divided into 100 minutes, and each minute further segmented into 100 seconds.
4. Radian measure: The radian (C) is defined as the measure of central angle subtended by an arc of length equal to the radius of the circle.
TEST YOURSELF
1. The degree measure corresponding to the radian measure 8 c π
is
(1) 22°30' (2) 26°30'
(3) 24°30' (4) 28°30'
2. The length of an arc of a circle of radius 5 cm subtending a central angle measuring 15° is
(1) 5 cm 12 π (2) 5 cm 6 π
(3) 7 cm 12 π (4) 7 cm 6 π
3. The angle subtended at the centre of a circle of diameter 50 cm by an arc of length 11 cm in degrees is
(1) 25°12' (2) 28°12'
(3) 23°12' (4) 25°18'
The relation between systems of measurement of angle is 2 90100 DGR π ==
• 180 1Radian =degree=5717'15'' ° π
• 1degreeradian = 0.0175 rad 180 = π
1. If the angles A, B, C of a triangle ABC are in arithmetic progression, then find the measure of angle B in radians.
Sol. Let A, B, C are angles in a triangle ABC, given that 2B = A + C and A + B + C = 180°
Hence, B = 60°.
The radian measure is 180 π times of 60°
It implies that 3 B π =
Try yourself:
1. Express 5 3 π in degrees.
Ans: 300°
4. The angle through which a pendulum swings, if its length is 50 cm and the tip describes an arc of length 10 cm, in degree, is
(1) 11°27' (2) 11°27'16'' (3) 12°27'16'' (4) 12°27'
5. A horse is tied to a post by a rope. The horse moves along a circular path of length 88 m, when it has traced out 72° at centre. Then the length of the rope is
(1) 70 m (2) 65 m
(3) 60 m (4) 75 m
6. If the degree measure corresponding to the radian measures 2 15 c π is k°, then k =______.
7. If the radian measures corresponding to the following degree measures 340° is kc m π
, then k + m = ______.
8. If the radian measure corresponding to the degree measure 75° is 12 kc π , then k = Answer Key
3.2 TRIGONOMETRIC RATIOS AND IDENTITIES
Trigonometric ratios are fundamental relationships between the angles and sides of triangles. Sine, cosine and tangent among others express these connections, aiding in the analysis of geometric configurations. These ratios play a pivotal role in diverse applications, from physics to engineering, unraveling the intricacies of spatial relationships.
3.2.1 Introduction to Trigonometric Ratios
A ray OP makes an angle q with x –axis as shown in the figure. Here, q is acute angle
Adjacent side cos Hypotensue x r
Opposite side tan
Adjacent side y x θ ==
Adjacent side cot
Opposite side x y
Hypotenuse sec
Adjacent side r x
Hypotenuse csc
Opposite side r y θ ==
Among the trigonometric ratios, sin q , csc q are reciprocals to each other, i.e., 1 sin;cos,sec csc = θθθ θ are reciprocals to each other, i.e., 1 cos sec θ θ = and tan q , cot q are reciprocals to each other, i.e., 1 tan cot θ θ =
Trigonometric Identities
1. sin2 x + cos2 x = 1 for all x ∈ R
2. 1 + tan2 x = sec2 x for all ()21, 2 xRnnZ π
3. 1 + cot2 x = csc2 x for all x ∈ R–{nπ, n ∈ Z}
Points to Remember:
1. The identity 1 + tan 2 x = sec 2 x can be written as sec2 x – tan 2 x = 1. It implies that (sec x – tan x ), (sec x + tan x ) are reciprocals to each other.
2. The identity 1 + cot 2 x = csc 2 x can be written as csc 2 x – cot 2 x = 1. It implies that (csc x – cot x ), (csc x + cot x ) are reciprocals to each other.
2 Prove that cos1cossin 1sin1cossin xxx xxx ++ = −+−
Sol Consider right hand side of the equation and multiply both numerator and denominator with 1 + cos x – sin x and then simplify
LHS: 1cossin 1cossin xx xx ++ = +− () ()
1cossin1cossin
RHS: 1cossin1cossin 1cossin 1cossin 1cos2cos1cos 21cos2sin1cos 2coscos1
RHS: 1cossin1cossin 1cossin 1cossin 1cos2cos1cos 21cos2sin1cos 2coscos1 cos 21cos1sin1sin xxxx xxxx
Therefore, cos1cossin 1sin1cossin xxx xxx ++ = −+−
Try yourself:
2. If 10 sin 4 x + 15 cos 4 x = 6 then find the value of 27 csc 6x + 8 sec 6x.
Ans: 250
3.2.2 Properties of Trigonometric Ratios
1. Transformation of trigonometric ratios in terms of other trigonometric ratios is represented in the following table.
2. Sign convention of trigonometric ratios in each quadrant: The sign of trigonometric ratios of angle q depends on the value of q.
i. If the angle q lies in the first quadrant
then all the trigonometric ratios are positive.
ii. If the angle q lies in the second quadrant
, then sin q , csc q are positive and all other trigonometric ratios are negative.
iii. If the angle q lies in the third quadrant
sine
cosine
tangent
cotangent
secant
cosecant
Increases from 0 to1
Decreases from 1 to 0
Increases from 0 to ∞
Decreases from ∞ to 0
Increases from 1 to ∞
Decreases from ∞ to 1
and all other trigonometric ratios are negative.
iv. If the angle q lies in the fourth quadrant 3
then cos q , sec q are positive and all other trigonometric ratios are negative.
3. Trigonometric ratios of Special angles:
4. The variations of trigonometric ratios in each quadrant are as shown in the table.
Decreases from 1 to 0
Decreases from 0 to –1
Decreases from 0 to –1
Increases from –1 to 0
Increases from –1 to 0 Increases from 0 to 1
Increases from – ∞ to 0 Increases from 0 to ∞ Increases from – ∞ to 0
Decreases from 0 to – ∞
Increases from – ∞ to –1
Increases from 1 to ∞
Decreases from ∞ to 0
Decreases from –1 to – ∞
Increases from – ∞ to –1
Decreases from 0 to – ∞
Decreases from ∞ to +1
Decreases from –1 to– ∞
3. If 1 cos 2 x =− and 2 x π <<π , then find the value of 4 tan2 x – 3 csc2 x.
Sol. Given that x lies in the second quadrant, and numerically, 1 cos 2 x =−
Since x lies in the second quadrant, tan x is negative and csc x is positive.
Hence,
Therefore, 4 tan2 x – 3 csc2 x = 8.
Try yourself:
3. Find cos x, if 26 sin 5 x =− and x lies in the third quadrant.
Ans: 1 5
3.2.3 Graphs of Trigonometric Functions
1. The graph of y = sin x
The domain of y = sin x is R and its range is [–1, 1].
2. The graph of y = cos x
The domain of y = cos x is R and its range is [–1, 1].
3. The graph of y = tan x y
x
The domain of y = tan x is ()21; 2 RnnZ π −+∈ and its range is (– ∞ , ∞ ). The function y = tan x is not defined at odd multiples of 2 π
4. The graph of y = csc x y
y = csc x
The domain of y = csc x is R–{nπ : n ∈ Z} and its range is (– ∞ , –1] ∪ [1, ∞ ). The function y = csc x is not defined at all multiples of π.
5. The graph of y = sec x
The domain of y = sec x is ()21; 2 RnnZ π −+∈
and its range is (–∞, –1]∪[1, ∞). The function y = sec x is not defined at odd multiples of 2 π
6. The graph of y = cot x
The domain of y = cot x is R–{nπ : n ∈ Z} and its range is (– ∞ , ∞ ). It is not defined at all multiples of π.
4. Draw the graph of 2cos 6 yx
Sol. Given: 2cos 6 yx
The transformation () cos 6 fxx
shifts the graph of y = cosx to 6 π units right.
=−
The transformation () 2cos 6 fxx
stretch vertically the graph of
to 2 times.
Therefore the graph of
Try yourself:
4. Draw the graph of π
=−
3.2.4 Values of Trigonometric Ratios of Allied Angles
Two angles are said to be allied when their sum or difference is either zero or a multiple of 2 π The angles allied to x are
and so on.
1. Trigonometric ratios of (– x):
The angle (–x) lies in the fourth quadrant. Cosine and secant functions are positive, remaining all trigonometric ratios are negative
i. sin(–x) = –sin x, csc(–x) = –csc x
ii. cos(–x) = cos x, sec(–x) = sec x
iii. tan(–x) = –tan x, cot(–x) = –cot x
Important Points:
■ If f (– x ) = f ( x ), then the function f ( x ) is called even function.
■ If f (– x ) = – f ( x ), then the function f ( x ) is called odd function.
■ The functions sin x , csc x , tan x , cot x are odd functions; cos x , sec x are even functions.
2. Trigonometric ratios of 2 x π :
Let x be an acute angle
The angle 2 x π lies in the first quadrant. In first quadrant all trigonometric ratios are positive.
i. sincos,cossin 22 xxxx
ii. cscsec,seccsc 22 xxxx ππ
iii. tancot,cottan 22 xxxx
3. Trigonometric ratios of 2 x π
+
:
The angle 2 x
lies in the second quadrant. In this quadrant, sine and cosecant functions are positive and the remaining trigonometric ratios are negative.
i. sincos,cossin 22 xxxx
ii. cscsec,seccsc 22 xxxx
iii. tancot,cottan 22 xxxx
4. Trigonometric ratios of (π – x):
The angle (π – x ) lies in the second quadrant. In this quadrant, sine, cosecant functions are positive and the remaining trigonometric ratios are negative.
i. sin(π – x) = sin x, cos (π – x) = –cos x
ii. csc(π – x) = csc x, sec (π – x) = –sec x
iii. tan(π – x) = –tan x, cot(π – x)= –cotx
5. Trigonometric ratios of (π + x):
The angle (π + x) lies in the third quadrant. In this quadrant tangent and cotangent functions are positive and the remaining trigonometric ratios are negati ve.
i. sin(π + x ) = –sin x , cos (π + x ) = –cos x
ii. csc(π + x ) = –csc x , sec (π + x ) = –sec x
iii. tan(π + x) = tan x, cot(π + x) = cot x
6. Trigonometric ratios of 3 2 x π :
The angle 3 2 x π lies in the third quadrant. In this quadrant, tangent and cotangent functions are positive and the remaining trigonometric ratios are negative.
i. 33 sin cos,cos sin 22 xxxx ππ −=−−=−
ii.
33 csc sec,sec csc 22 xxxx ππ
iii. 33 tancot,cottan 22 xxxx ππ
−=−=
7. Trigonometric ratios of 3 2 x π + :
The angle 3 2 x π +
lies in the fourth quadrant. In this quadrant, cosine and secant functions are positive and the remaining all trigonometric ratios are negative.
i.
33 sin cos,cossin 22 xxxx ππ
ii. 33 csc sec,seccsc 22 xxxx
iii. 33 tan cot,cot tan 22 xxxx ππ
8. Trigonometric ratios of (2π – x):
The angle (2π – x ) lies in the fourth quadrant. In this quadrant, cosine and secant functions are positive and the remaining all trigonometric ratios are negative.
i. sin(2π – x) = –sin x, cos(2π – x) = cos x
ii. csc(2π – x) = –csc x, sec(2π – x) = sec x
iii. tan(2π – x) = –tan x, cot(2π – x) = –cot x
5. Find the value of csc (390°).
Sol. Express 390° = 4.90° + 30°. It lies in the first quadrant.
Since the number 4 is even, there is no change in the trigonometric function.
Hence, csc(390°) = csc(30°) = 2
Try yourself:
5. Find the value of 11 sin 3 π
6. If a cos q + b sin q = a and a sin q≠ 0 then a sin q – b cos q = b
7. If a cos q + b sin q = b and a cos q≠ 0 then b cos q – a sin q = a
8. If a sec q + b tan q = c and a tan q + b sec q = k then a2 – b2 = c2 – k2
9. If a csc q + b cot q = c and a cot q + b csc q = k, then a2 – b2 = c2 – k2
10. cot2 q – cos2 q = cot2 q cos2 q , tan2 q – sin2 q = tan2q sin2 q , and csc2 q + sec2 q = csc2 q sec2 q.
11. θθθθθ +=−=− 44222 1 sincos12sincos1sin2 2
θθθθθ +=−=− 44222 1 sincos12sincos1sin2 2
. Ans: 3 2
3.2.5 Standard Results on Trigonometric Ratios
1. For any x ∈ R,
12. 66222 3 sincos13sincos1sin2 4 θθθθθ +=−=−
66222 3 sincos13sincos1sin2 4 θθθθθ +=−=−
i. sin2 x + csc2 x ≥ 2
ii. cos2 x + sec2 x ≥ 2
iii. tan2 x + cot2 x ≥ 2
2. If n is odd, then
i. sin x + sin(π+ x) + sin(2π + x)+ ......... + sin(nπ + x) = 0
ii. cos x + cos(π+ x) + cos(2π + x)+ ....... + cos(nπ + x) = 0
3. If n is even then
i. sin x + sin(π+ x) + sin(2π + x)+ ......... + sin(nπ + x) = sin x
ii. cos x + cos(π+ x) + cos(2π + x)+ ....... + cos(nπ + x) = cos x
4. If a cos q + b sin q = c then
222 sincos ababc θθ−=±+−
5. If a cos q – b sin q = c then
222 sincos ababc θθ+=±+−
13. sin2 q + cos4 q = cos2 q + sin4 q = 1– sin2 q cos2 q
14. If P n = cos n q + sin n q, then 22 2 2 sincos nn n PP P +θθ =
6. Prove that (sinx + cosecx)2 + (cosx + secx)2 = tan2x + cot2x + 7
Sol. (sinx + cosecx)2 + (cosx + secx)2 = sin2x + coec2x + 2 + cos2x + sec2x + 2 = 1 + 4 + 1 + cot2x + 1 + tan2x = tan2x + cot2x + 7
Try yourself:
6. Find the value of (1+ cot x – cosec x ) (1+ tanx + sec x).
Ans: 2
TEST YOURSELF
1. If q lies in the first quadrant and 5 tan q = 4, then 5sin3cos sin2cos θθ θθ = + (1) 5/14 (2) 3/14 (3) 1/14 (4) 0
2. If tan q= 1 4 p p , then sec q- tan q=
(1) () 1 2or 2 p p (2) () 1 or2 2 p p
(3) 1 (or) 2 2 p p (4) () 1 or2 2 p p
3. If sin2q = K2 (0<k<1) and 180° < q<270°, then θθ +−= 2 2 cot 1.sec 1 K K K
(1) 2 (2) –2 (3) 1 (4) 0
4 If 2 sin x + 5 cos y + 7 sin z = 14, then 7tan4 cos 6 cos 2 x yz +−= ______.
5. If 3 cot 4 A = and 2 cos 3 sin 4 cos sin AA AA + = k, then 8k is __________.
6. The graph y = cosx is decreasing in
(1) [0, π ] (2) [– π , 0 ] (3) [ π , 2 π ] (4) none
7. Sin 4530° = (1) 1 2 (2)
(3) 3 2 (4)
8. If tan 35° = k, then the value of 00 00 tan145tan125 1tan145tan125 = + (1) 2 2 1 k k (2) 2 2 1 k k + (3) 2 1 2 k k (4) 2 2 1 1 k k +
9. () 3 2 1 cos21 12 k k = π ∑−= (1) 0 (2) 1 2 (3) 1 2 (4) 3 2
10. log tan1° + logtan2° + ....+ log tan 89° = (1) 1 (2) 0 (3) –1 (4) 2
11. cos2 5° + cos210° + cos215° + .....cos2360° = (1) 18 (2) 27 (3) 36 (4) 45
12. ++++=2222 sin5sin10sin15...sin180 oooo
13. = cos1 cos2 cos3............cos179 oooo
14. If θθθπ +++…∞=+<< 2 1 sinsin423, 0 and , 2 θπ ≠ then q (1) , 63 ππ (2) 5 , 36 ππ (3) 2 , 63 ππ (4) 2 , 33 ππ
15. If 15 sin4a + 10 cos4a = 6, for some R α ∈ then the value of αα + 66 27 sec 8 cosec is equal to _____.
16. If 1+ 4 tan q = 4 sec q, then 8 17 cosθ −=__.
17. Let f and g be function defined by f ( q ) = cos2 q and g( q ) = tan2q. Suppose a and b satisfy 2f( a ) – g( b ) = 1. Then the value of 2f( b ) – g( a ) is __.
Answer Key
(1) 1 (2) 2 (3) 4 (4) 11 (5) 18 (6) 1 (7) 2 (8) 3 (9) 4 (10) 2 (11) 3 (12) 18 (13) 0 (14) 4 (15) 250 (16) 0 (17) 1
3.3 COMPOUND ANGLES
The algebraic sum of two or more angles is called a compound angle.
3.3.1 Trigonometric Ratios of Compound Angles:
1. Trigonometric ratios of sum or difference of two angles:
For any A, B ∈R, the trigonometric ratios of A + B and A – B
i. () Sinsincoscossin ABABAB +=+ ii. () Sinsincoscossin ABABAB −=−
iii. () coscoscossinsin ABABAB +=−
iv. () coscoscossinsin ABABAB −=+
2. Sum or difference of trigonometric ratios of compound angles
i. ()() sinsin2sincos ABABAB ++−=
ii. ()() sinsin2cossin ABABAB +−−=
iii. ()() coscos2coscos ABABAB ++−=
iv. ()() coscos2sinsin ABABAB +−−=−
3. Suppose that A,B, A + B are not multiples of 2 π . Then () tantan tan 1tantan AB AB AB + +=
4. Suppose that A, B, A – B are not multiples of 2 π . Then () tantan tan 1tantan AB AB AB −= +
5. Suppose that A, B, A + B are not multiples of π. Then ()+= + cotcot1 cot cotcot AB AB BA
6. Suppose that A, B, A – B are not multiples of π. Then ()+ −= cotcot1 cot cotcot AB AB BA
7. ()() 22 22 sinsin sinsin coscos AB ABAB BA +⋅−=
8. ()() 22 22 cossin coscos cossin AB ABAB BA +⋅−=
9. Trigonometric ratios of sum of three angles:
For any A, B, C ∈ R, i. sin(A + B + C) = ∑ sinA cosB cosC – sinA sinB sinC ii. cos(A + B + C) = cosA cosB cosC – ∑ sinA sinB cosC For suitable angles of A, B, C
i. () tantan tan 1tantan AA ABC
ii. () cotcot cot 1cotcot AA ABC AB ∑−∏ ++=
7. In a triangle ABC, 2 cos 3 A =− , find the quadratic equation whose roots are sin A , tan A.
Sol. Given: 2 cos 3 A =− , Hence, the angle A is obtuse angle.
So, 5 sin 3 A = and 5 tan 2 A =− Equation, having roots
be taken as
This can be simplified as 2 65550 xx−+=
Try yourself:
7. Expand cos(A – B – C). Ans: cos A cos B cos C + sin A sin B cos C+ sin A cos B sin C – cosA sin B sin C
3.3.2
Results on Compound Angles
1. Trigonometric ratios of 15°, 75°: Angle q 15° 75°
2. Trigonometric ratios of 11 22,67 22 °° :
3. Some important results:
i. sin q .sin(60– q ).sin(60+ q ) ()() 1 sinsin60sin60sin3 4 θθθθ ⋅−⋅+=
ii. cos q .cos(60– q ).cos(60+ q )()() 1 coscos60cos60cos3 4 θθθθ ⋅−⋅+=
iii. sin q +sin( q +120°)+sin( q –120°) = 0
iv. sin q +sin( q +240°)+sin( q –240°) = 0
v. cos q +cos( q +120°)+cos( q –120°) = 0
vi. cos q +cos( q +240°)+cos( q –240°) = 0
4. If A + B = 45° or A + B = 225°, then
i. (1 + tan A)(1 + tan B) = 2
ii. (1 – cot A)(1 – cot B) = 2
iii. (1 + cot A)(1 + cot B) = 2 cotA cotB
iv. 1 tan2221 2 °=−
5. If A + B = 135° or A + B = 315°, then
i. (1 – tan A)(1 – tan B) = 2
ii. (1 + cot A)(1 + cot B) = 2
iii. (1 + tan A)(1 + tan B) = 2 tanA tanB
iv. 1 tan6721 2 °=+
6. If A + B + C is a multiple of π, then
i. tanA + tanB + tanC = tanA tanB tanC
ii. cotA cotB + cotB cotC + cotC cotA =1
7. If A + B + C is an odd multiple of 2 π , then
i. tan A tan B + tan B tan C + tan C tan A = 1
ii. cotA + cotB + cotC = cotA cotB cotC
8. If 2 ABC π ++= , then find the value of () cos coscos BC BC ∑
Sol. () cos cosB cosC cosB cosCsinB sinC cosB cosC 1tantan 3tantantantantantan 314 BC BC ABBCCA +
Try yourself:
8. If in a triangle ABC, the value of tan A tan B tan C is 6, and tan A tan B = 2, then find the values of tan A, tan B, tan C.
Ans: tanA =1, tanB = 2, tanC = 3
TEST YOURSELF
1. = + tan225cot81.cot69 cot261tan21 (1) 1 (2) 1 2 (3) 3 (4) 1 3
2. If 1 sin ,in ,0A, B 4 1 105 AsB=<< π = , then A + B = (1) π 2 (2) π 3 (3) π 4 (4) π 5
3. If θ=φ= tanandtan11 23 , then the value of q+φ is (1) π 6 (2) π (3) 0 (4) π 4
4. If π ++= , 2 ABC then () + ∑= cos coscos BC BC (1) 1 (2) 2 (3) 3 (4) 4
5. If == 37 cos, sin 525 AB and 90° < A < 180°, 0° < B < 90°, then 4|tan(A + B)| =
6. If A + B + C = 180°, then + ∑= + cotcot tantan AB AB
7.
3 tan.tan 44 (1) 0 (2) –1 (3) 1 (4) 2
8. If tan 8A – tan 5A– tan 3A = k, tan 8A tan 5A.tan 3A, then k = (1) 1 (2) 2 (3) 3 (4) 4
9. If A and B are actue angles satisfying sin A = sin2B and 2cos2A = 3cos2B, then A + B = (1) 60° (2) 75° (3) 80° (4) 90°
10. (1 + cot 78°)(1 + cot 57°) = (1) 0 (2) 1 (3) 2 (4) 1 2
11. In triangle ABC, if tanA + tanB + tanC = 6 and tanA tanB = 2, then the values of tanA, tanB and tanC are, respectively, (1) 1, 2, 3 (2) 3, 2/3, 7 (3) 4, 1/2, 3/2 (4) 5, 2/5, 3/5
12. If tan b = 2sin a sin γ cosec( a + γ ), then cot a , cot b , and cot γ are in (1) AP (2) GP (3) HP (4) AGP
13. If 2 α+β+γ=π and cot a , cot b , and cot γ are in AP, then cot a , cot γ = ___.
14. If () ()+α+α=α∈ π 1tan 1tan42, 0, 10 , then 20
15. sin10° – sin110° + sin130° = ___.
Answer Key (1) 3 (2) 3 (3) 4 (4) 2 (5) 3 (6) 1 (7) 2 (8) 1 (9) 2 (10) 3 (11) 1 (12) 1 (13) 3 (14) 1 (15) 0
3.4 MULTIPLE AND SUBMULTIPLE ANGLES
Let A be an angle. The measures of angles 2 A , 3 A , 4 A are called multiple angles and the measures of angles ,,,... 234 AAA are called submultiple angles.
3.4.1 Trigonometric Ratios of Multiple and Submultiple Angles
1. For any A ∈ R, the trigonometric ratios of 2A
i. sin(2A) = 2 sinA cosA
ii. () 22 2 2 cossin cos212sin 2cos1 AA AA A
iii. () 2 2tan tan2 1tan A A A =
Here, A, 2A are not odd multiples of 2 π
iv. () 2 cot1 2cot cot2 1 cottan 2 A AA AA
= Here, A , 2 A are neither integral multiples nor odd multiples of 2 π
2. Replace A with 2 A in the above relations to get the following.
i.
sin2sincos 22 AA A
=
3: Trigonometric Functions
ii. 22 2 2 cossin 22 cos12sin 2 2cos1 2 AA A A A
=−
iii. 2 2cos1cos 2 A A
iv. 22 coscossin 22 AA A =−
v. 2 2sin1cos 2 A A
3. 2 2tan 2 tan 1tan 2 A A A = Here , 2 A A are not odd multiples of 2 π
4. 2 cot1 2 cot 2cot 2 A A A = Here , 2 A A are not odd multiples of 2 π
5. 2cotcottan 22 AA A =−
Here A is not an integral multiple of π and 2 A is neither an integral multiple nor an odd multiple of 2 π
6. Express sin2A, cos2A, tan2A in terms of tan A:
Suppose that the measure of angle A is not an odd multiple of 2 π , then
i. 2 2tan sin2 1tan A A A = +
ii. 2 2 1tan cos2 1tan A A A = +
iii. 2 2tan tan2 1tan A A A =
Here 2A is not odd multiples of 2 π
7. Substitute 2 A in place of A in the above formulae
i. 2 2tan 2 sin 1tan 2 A A A
=
+
Here, A is not an odd multiple of π.
ii. 2 2 1tan 2 cos 1tan 2 A A A
=
+
Here, A is not an odd multiple of π.
iii. 2 2tan 2 tan 1tan 2 A A A
Here A is not an odd multiple of , 2 π π
8. Trigonometric ratios of triple angles: For any measure of angle A ∈ R, i. sin 3A = 3 sinA – 4 sin3A ii. cos 3A = 4 cos3A – 3 cosA
iii. 3 2 3tantan tan3 13tan AA A A = 3A, A are not odd multiples of 2 π
iv. 3 2 3cotcot cot3 13cot AA A A = 3A, A are not multiples of π
9. Find the value of 22 22 sin3c 3 os sincos AA AA
Sol. LHS: 22 22 sin3c 3 os sincos AA AA
sin2sin4
sincos
2sincos2sin2cos2 sincos
4sincos2sincoscos2
sincos
8cos2
Try yourself:
9. Find the value of
Ans: cosec
3.4.2 Expressing the Trigonometric Ratios in Terms of cos 2A
We can express all trigonometric ratios in terms of cos 2A.
i. For all A ∈ R, 1cos2 sin 2 A A =±
ii. For all A ∈ R, 1cos2 cos 2 A A + =±
iii. For all ()21;, 2 ARnnZ π ∈−+∈ 1cos2 tan 1cos2 A A A =± +
iv. For all {} ; ARnnZ π ∈−∈ , 1cos2 cot 1cos2 A A A + =±
v. For all ()21; 2 ARnnZ π ∈−+∈ , 2 sec 1cos2 A A =± +
The sign (±) of the above trigonometric ratios are determined depending on the quadrant in which the angle A lies.
Substitute 2 A for A in the above results to get the following relations.
i. 1cos sin 22 AA =± for all A ∈ R
ii. 1cos cos 22 AA + =± for all A ∈ R
iii. 1cos tan 21cos AA A =± + for all
() {} 21; ARnnZ ∈−+π∈
The sign (±) of the above trigonometric ratios are determined depending on the quadrant in which the angle 2 A lies.
10. If π < x < 2π, then find 1cos 1cos x x + .
Sol. 2 2 2 22 2cos 1cos 2 1cos 2sin 2 cos 2 cot 2 sin 2 x x x x xx x x x π πππ <<⇒<< + = ==−
Try yourself:
10. If π < x < 2π and 1cos1cos 1cos1cos xxk xx +− += −+ cosecx, then find the value of k. Ans: –2
3.4.3 Finding the Trigonometric Ratios of Special Angles
To find the trigonometric ratios of special angles, use trigonometric ratios of submultiple angles.
The trigonometric ratios of 18°:
i. 51 sin18 4 °=
ii. 1025 cos18 4 °=+
The trigonometric ratios of 1 22 2 ° :
Substitute 1 22 2 ° in the formula of all trigonometric ratios in terms of sin,cos,tan 222 AAA to get the values of trigonometric ratios of 1 22 2 °
i. 121 sin22 2 22 °=
ii. 121 cos22 2 22 °=+
iii. 1 tan2221 2 °=−
iv. 1 cot2221 2 °=+
The trigonometric ratios of 36°:
Use the values of sin18°, cos18° to get sin36°, cos36°.
i. 1025 sin36 4 °=
ii. 51 cos36 4 °=+
The trigonometric ratios of 72°:
Use the values of sin18°, cos18° to get sin72°, cos72°
i. 1025 sin72 4 °=+
ii. 51 cos72 4 °=
11. Find the value sin2 24° – sin2 6°.
Sol. sin(24° + 6°) sin(24° – 6°) sin30° sin 18° sinsin 610 15151 248 =×= ππ
Try yourself:
11. Find the value of 22 2 sinsin 55 ππ
Ans: 5 4
3.4.4 Expressing the Trigonometric Ratios of 2 A in Terms of sin A
1. Use the trigonometric ratios of multiple and submultiple angles formula to get the trigonometric ratios of 2 A in terms of sinA.
i. ()()2cos1si s n n 2 1i A A A =±±− ++
ii. ()()2sin1si s n n 2 1i A A A =±±− +−
iii. 11tan2 tan 2tan AA A −±+ =
Let cos,sin 22 AA CS==
2. If , 244 A ππ ∈− then C + S > 0, C – S > 0
i. 1sin1sin2cos 2 A AA ++−=
ii. 1sin1sin2sin 2 A AA +−−=
3. If 3 , 244 A ππ ∈
then C + S > 0, C – S < 0
i. 1sin1sin2sin 2 A AA ++−=
ii. 1sin1sin2cos 2 A AA +−−=
4. If 35 , 244 A ππ ∈
then C + S < 0, C– S < 0
i. 1sin1sin2cos 2 A AA ++−=−
ii. 1sin1sin2sin 2 A AA +−−=−
5. If 57 , 244 A ππ ∈
then C + S < 0, C – S > 0
i. 1sin1sin2cos 2 A AA ++−=−
ii. 1sin1sin2sin 2 A AA +−−=−
This can be represented as below.
Trigonometric ratios of 1 9,7 2
The values of sin9°, cos9° are as below.
i. 3555 sin9 4 °=+−−
ii. 3555 cos9 4 °=++−
The values of 11 cot7,tan7 22 °° are as below.
i. 1 cot76432 2 °=+++
ii. 1 tan76432 2 °=−−+
12. Find the value of tan 9° – tan 27° – tan 63° + tan 81°.
Sol. (tan 9° + cot 9°) – (tan 27° + cot 27°) 11 sin9cos9sin27cos27 2222 sin18sin54sin18cos36 88 4 5151 =−
Try yourself:
12. Find the value of tan 6° tan 42° tan 66° tan 78°.
Ans: 1
3.4.5 Results on Multiple and Submultiple Angles
1. If a1, a2, .......a n are in arithmetic progression with common difference d then i. sin a1 + sin a2 + …+ sin a n = 1 sinsin 22 sin 2 n aand d +
ii. cos a1 + cos a2 + …+ cos a n = 1 cossin 22 sin 2 n aand d +
2. When q is not multiple of π and not multiple of 2 π :
i. () tan1sectan 2 θθθ +=
ii. () ()()tan1sec1sec2...1sec2 2 tan2 n n θθθθ θ +++ = (1+ sec q )(1+ sec2 q ).... (1+ sec2nq ) = tan2nq
iii. (2 cos q – 1)(2 cos q +1) = 2cos 2 q +1
iv. () () () () 1 2cos12cos212cos41.... 2cos21 2cos21 2cos1 n n θθθ θθ θ + −= +
3. For any q∈ R,
i. cosq + cos(120 + q) + cos(120 – q) = 0
ii. sinq + sin(120 + q) – sin(120 – q) = 0
iii.
iv.
v.
vii.
i. ()()222 3 sinsinsin 2 θαθαθ +−++= ii. ()()222 3 coscoscos 2 θαθαθ +−++=
iii. ()() 1 sinsinsinsin3 4 θαθαθθ +−=
iv. ()() 1 coscoscoscos3 4 θθαθαθ +−=
v. tan q tan( q + a ) tan ( q – a ) = tan 3 q
5. When q , 3 q are not odd multiples of 2 π
i. tan q + tan(60° + q ) + tan(120° + q ) = 3tan 3 q
ii. tan q + tan(120° + q ) + tan(240° + q ) = 3tan 3 q
iii. tan q + tan(240° + q ) + tan(300° + q ) = 3tan 3 q
iv. tan q + tan(60° + q ) + tan(300° + q ) = 3tan 3 q
6. When q is not a multiples of π: i. cos q cos2 q cos4 q .....cos(2n–1q ) ()() 1 sin2 coscos2cos4...cos2 2sin n n n θ θθθθ θ =
ii. If 21 n θπ = + then cos q cos 2 q cos 4 q .... () 1 1 cos2 2 n n θ=
viii. ()() 33
coscos120cos120 3 cos3 4 +°−+°+ = θθθ θ
ix. ()() 33 3 coscos240cos240 3 cos3 4 +°−+°+ = θθθ θ
x. ()() 33 3 coscos300cos300 3 cos3 4 −°−−°+ = θθθ θ
4. If a = 60°, 120°, 240°, 300° then
iii. If 21 n θπ = then cos q cos 2q cos 4q.... () 1 1 cos2 2 n n θ=−
13. Find the value of 24 coscoscos 777 πππ Sol. 3 3 8 sin sin2 7 2sin 8sin 7 sin sin 1 7 7 8
Try yourself:
13. Find the value of 57 sinsinsin 181818 πππ Ans: 1/8
TEST YOURSELF
1. 22 22 tan2tan 1tan2tan θθ θθ =
(1) tan 3 q cot q (2) cot 3 q tan q
(3) cot 3 q cot q (4) tan 3 q tan q
2. π θθ <+++= If then 2222cos8 16
(1) 1 8 (2) 2 cos q (3) 2cos 2 θ (4) 2cos 4 θ
3. If A is not an integral multiple of π/2, then cosec 2A + cot 2A=
(1) tan A
(2) cot A + 2cot 2A
(3) tan A + 2cot 2A (4) tan 2A
4. += cot tan cot cot 3tan tan 3 xx xxxx
5.
, then tan 3x is equal to ____.
6. 2234 coscos 55 += ππ (1) 4 5 (2) 5 2 (3) 5 4 (4) 3 4
7. sin6°sin42°sin66°sin78° = (1) 1 2 (2) 1 4 (3) 1 8 (4) 1 16
8. If
then
9. If A – B = 60°, then sin2A + sin2B – sinAsinB = (1) 1 2 (2) 3 4 (3) 1 (4) 3 2 10. cos3110° + cos310° + cos3130° = (1) 3 8 (2) 3 4 (3) 33 8 (4) 33 4 11. 4 cos312° – 3 sin78° = (1) 51 4 (2) + 51 4 (3) 1025 4 (4) + 1025 4
12. 12 sinsin 1010 ππ
=
(1) 1 (2) 1 2 (3) 1 2 (4) 1 4
13. If cos3A + cos3(120° + A) + cos3(120° – A) = 4 k cos3A, then k =______.
14. cos 2( a + b ) + cos 2( a – b ) – cos2 a cos2 b =_________.
15. If cos236° + cos272° = k, then 4k =_____.
Answer Key (1) 4 (2) 2 (3) 3 (4) 1 (5) 1 (6) 4 (7) 4 (8) 3 (9) 2 (10) 3 (11) 2 (12) 4 (13) 3 (14) 1 (15) 3
3.5 TRANSFORMATIONS
In this part, the sum or the difference of two trigonometric ratios transforms into products and vice versa.
1. Transformation of sum or difference into product of trigonometric functions: For all C, D ∈ R, i. sinsin2sincos 22
CHAPTER 3: Trigonometric Functions
CDCD CD +−
ii. sinsin2cossin 22
CDCD CD +−
iii. coscos2coscos 22
CDCD CD +−
iv. coscos2sinsin 22
2. Transformation of product into sum or difference of trigonometric functions
For all A, B ∈ R,
i. 2sinA cosB = sin(A + B) + sin (A – B)
ii. 2cosA sinB = sin(A + B) – sin (A – B)
iii. 2cosA cosB = cos(A + B) + cos (A – B)
iv. –2sinA sinB = cos(A + B) – cos (A – B)
Results on transformations:
1. In a triangle ABC, A + B + C = 180°.
i. sin2A + sin2B + sin2C = 4 sinA sinB sinC
ii. sin2A + sin2B – sin2C = 4 cosA cosB sinC
iii. sin2A – sin2B + sin2C = 4 cosA sinB cosC
iv. cos2 A + cos2 B +cos2 C = –1–4 cos A cosB cosC
v. cos2 A + cos2 B – cos2 C = 1–4 sin A sinB cosC
2. If cos x + cos y = a and sinx + siny = b, then
i. tan 2 xyb a + =
ii. () 22 2 sin ab xyab += +
iii. () 22 22 cos ab xyab += +
iv. () 22 2 tan ab xyab +=
3. If cos x – cos y = a and sin x – sin y = b, then
i. tan 2 xya b + =−
ii. () 22 2 sin ab xyab +=− +
iii. () 22 22 cos ba xyab += +
iv. () 22 2 tan ab xyab +=−
4. If cos x – cos y = a and sin x + sin y = b, then
i. tan 2 xya b =−
ii. () 22 2 sin ab xyab −=− +
iii. () 22 22 cos ba xyab −= +
iv. () 22 2 tan ab xyba −=−
5. If cos x + cos y = a and sin x – sin y = b , then
i. tan 2 xyb a =
ii. () 22 2 sin ab xyab −= +
iii. () 22 22 cos ab xyab −= +
iv. () 22 2 tan ab xyab −=
6. If sin (y + z – x), sin (z + x – y), sin (x + y –z) are in arithmetic progression then tanx, tany, tanz are in arithmetic progression.
14. Solve cos4cos3cos2 sin4sin3sin2 xxx xxx ++ ++
Sol. 4242
2coscos cos3 22 4242
2sincos sin3 22
2cos3coscos3
2sin3cossin3
cos3(2cos1)
sin3(2cos1) cos3 cot3 sin3 xxxx x xxxx x xxx xxx xx xx x x x
Try yourself:
14. Find the value of cos20° + cos 100° + cos140°. Ans: 0
TEST YOURSELF
1. sin47° + sin61° – sin11° – sin25° = (1) sin 7° (2) cos 7° (3) tan 7° (4) sin 14°
2. If +=+= 11 sinsin,coscos 43xyxy , then tan 2 xy + = (1) 1 4 (2) 1 2 (3) 3 4 (4) none
3. If a+b=γ, then cos2a + cos2b + cos2γ – 2 cos a cos b cos γ = (1) 1 (2) 0 (3) –1 (4) –2
4. The value of ()() () cos cos 2cos ABCABC BC +++−− = + (1) cosA (2) sinA (3) 2cosA (4) 2sinA
(4) 4sincoscos 222 α+ββ+γγ−α
6. () () () () coscos3sin8sin2 sin5sincos4cos6 α−αα+α = α−αα−α (1) 2 (2) 3 (3) 1 (4) 4
7. 2 cos208sin10sin50sin70 sin80 °+°°° = ° _____.
8. If 3 sina = 5 sinb, then tan 2 tan 2 α+β
Answer Key (1) 2 (2) 3 (3) 1 (4) 1 (5) 2 (6) 3 (7) 2 (8) 4
3.6 PERIODICITY AND EXTREME VALUES
5. If α, β, and γ are any 3 angles, then cos a + cos b – cos γ – cos( a + b + γ ) = (1) 4coscoscos 222 α+ββ+γγ+α (2) 4cossinsin 222 α+ββ+γγ+α (3) 4coscoscos 222 α+ββ−γγ−α
Periodicity denotes repeating function values at regular intervals, crucial in trigonometric functions like sine and cosine. It this topic we will evalute extreme values of trigonometric functions.
3.6.1 Periodicity and Periodic Functions
A function f: A → B is said to be periodic function if there exists a real number p such that f(x + p) = f(x), ∀ x ∈ A. The smallest such value of p is called period of f or fundamental period of f.
If the period of a function f(x) is p, then f(x + p) = f(x) for all x in the domain of f(x). If the period of a function f ( x ) is p, then f ( x + np ) = f ( x ) for all x in the domain of f ( x ), where n is an integer. Every constant function is a periodic function which has no fundamental period.
Period of Trigonometric Functions:
1. The fundamental period of sin x , cos x , sec x, csc x is 2π.
2. The fundamental period of tan x, cot x is π.
3. Period of special functions:
i. The period of sin(ax + b), cos(ax + b) is 2 a π
ii. The period of |sin(ax + b)|, |cos(ax + b)| is a π
iii. The period of sinnx, cosnx, secnx, cscnx is π when n is even.
iv. The period of sinnx, cosnx, secnx, cscnx is 2π when n is odd.
v. The period of tann x, cotn x is π.
vi. The period of modulus of all six trigono-metric functions is π.
vii. The period of x –[x] is 1 and the period of kx– [kx] is 1 k .
viii. The period of {x} is 1.
ix. The algebraic functions, 23 ,,, xxxx etc are not periodic functions.
4. Periods of some more functions
i. Period of modulus of trigonometric functions is π.
ii. Period of a|sin x| + b|cosx| and a|csc x| + b|sec x| is 2 π if a = b and it is π when a ≠ b.
iii. Period of |sin x + cos x | and |sin x –cosx| is π
iv. Period of |tan x + cot x| and |tan x – cot x| is 2 π
v. Period of sin2n x + cos2n x, sec2n x + csc2n x and tan2n x + cot2n x is 2 π
15. What is the period of 2sin3cos 2 4tan27sin3 x x xx +
Sol. sinx period is 2π 2 cosperiodis4 1 2 2 x = π π tan2periodis 2 2 sin3periodis 3 2 LCMof2,4,,is4 23 x x
Try yourself:
15. What is the sine function whose period is 2 3 Ans: sin (3πx + b)
3.6.2 Extreme Values of Trigonometric Functions
1. Extreme values of standard trigonometric functions:
i. The maximum and minimum values of both sin x, cos x are 1, –1 on set of all real numbers, these values are called extreme values.
ii. There are no extreme values to the other four trigonometric functions.
iii. The extreme values of k sin x, k cos x are k, –k.
2. Extreme values of a cosx + b sin x + c:
i. The extreme values of a cosx + b sin x are 2222 , abab +−+
ii. The extreme values of a cosx + b sin x + c are 2222 , cabcab ++−+
iii. The extreme values of a cos(px + q) + b sin (px + q) + c are 2222 , cabcab ++−+
16. Find the extreme values of sin 2x – cos 2x over R
Sol. The extreme values of a cosx + b sin x + c are 2222 , cabcab ++−+
The extreme values of sin 2 x – cos 2x are 011,011 2,2 −+++ =−
Try yourself:
16. Find the maximum and minimum value of 7cos x – 24 sin x + 5 over R
Ans: 30, –20
3. Extreme values of a sin2 x + b cos x + c or a cos2 x + b sin x + c:
i. Reduce the given quadratic equation in terms of either sin x or cos x
ii. Write the equation in completing square form, and then write the extreme values.
17. What are the extreme values of sin2 x + 4 cos x + 7?
Sol. sin2 x + 4 cos x + 7
= 1– cos2x + 4 cosx + 7 = 12 – (cos2x – 4cosx + 4) = 12 – (cosx – 2)2
Minimum value = 12 – 9 = 3
Maximum value = 12 – 1 = 11
Try yourself:
17. What are the extreme values of 3 sin2x + 4?
Ans: 4, 7
4. The extreme values of a sin2 x + b sin x cos x + c cos2 x are ()2 2 22 acbac ++− ±
5. a2 sec2 x + b2 cosec2 x ≥ (a + b)2 occurs at 1 tan b x a
6. a2 sin2 q + b2 csc2 q≥2 ab, a2 cos2 q + b2 sec2 q≥2ab and a2 tan2 q + b2 cot2 q≥2ab
7. For the function 22222222 cossinsincos, axbxaxbx +++ minimum value is a + b at x = 0 and the maximum value is () 22 2 ab + and occurs at 4 x π =
8. Extreme values of cossin k axbxc ++
Case (i) : If the range of a cos x + b sin x + c is [ p, q ], p < q when both p, q are positive. In this c ase the extreme values of cossin k axbxc ++ are , kk qp
Case (ii): If the range of a cos x + b sin x + c is [ p, q ], p < q , pq < 0. In this case, the extreme values of the function cossin k axbxc ++ has neither maximum nor minimum values.
9. In a triangle ABC, the maximum value of i. sin A + sin B + sin C is 33 2
ii. cos A + cos B + cos C is 3 2
iii. sinsinsin 222 ABC is 1/8
iv. coscoscos 222 ABC is 33 8
v. cos A cos B cos C is 1 8
10. In a triangle ABC, the minimum value of
i. tan A + tan B + tan C is 33
ii. tan A tan B tan C is 33
iii. 222 tantantan 222 ABC ++ is 1
iv. 2 sinsin1 3 sin AA A ++ ≥
11. If range of f ( x ) is [– a , 0], then range of () 1 fx is 1 , a
18. Find the range of 1 5sin12cos13 xx+−
Sol. f(x) = 5 sinx + 12 cosx – 13
Minimum value of f(x) is 22 131326 cab−+=−−=−
Maximum value of f(x) is 22 13130 cab++=−+=
The range of f(x) is [–26, 0]
∴ The range of 1 ()fx is 1 , 26 −∞
Try yourself:
18. Find the range of 1 3sin4cos5 xx−+ Ans: 1 , 10
TEST YOURSELF
1. The period of sin2costan 234 xxx
+−
(1) 4 (2) 6 (3) 12 (4) 24
2. The period of |sinx + cosx| is (1) 2 π (2) π (3) 4 π (4) 2 π
3. The period of sin6x + cos6x is (1) 3 2 π (2) 2 π (3) π (4) 2 π
4. The sine function whose period is 3 is (1) 2 sin 3 x π
(3) 2 sin 3 x π
±
(4) sin 3 x
5. The period of |cot x | + |cos x | + |tan x | + |sinx| is (1) π (2) 2 π (3) 2 π (4) 4 π
6. The period of cosxcos(120° – x)cos(120° + x) is (1) 2 3 π (2) 3 π (3) π (4) 2 π
7. If () 22 sinsin, 8282 fxxx ππ
then the period of f is (1) π (2) 2 π (3) 3 π (4) 2 π
8. The range of 13cos33sin4 xx+− is
(1) [–18, 10] (2) [10, 18] (3) (–18, 10) (4) –18, 10
9. The minimum value of 27tan2θ + 3cot2θ is (1) 15 (2) 18 (3) 24 (4) 30
10. The minimum value of 2cosx – 3cos2x + 5 is
(1) –1 (2) 0 (3) 1 (4) 2
11. The range of cos2x + sin4x is (1) 1 ,1 2
(3) 3 ,2 2
(2) 3 1, 2
(4) 3 ,1 4
12. The maximum value of 3 5sin12cos19 xx−+ is (1) 1 (2) 1 2 (3) 1 3 (4) 1 4 Answer Key (1) 3 (2) 2 (3) 2 (4) 3 (5) 2 (6) 1 (7) 4 (8) 1 (9) 2 (10) 2 (11) 4 (12) 2
CHAPTER REVIEW
Measurement of Angles
1. The relation between systems of measurement of angle is 2 90100 DGR π ==
• 180 1Radian =degree=5717'15'' ° π
• 1degreeradian = 0.0175 rad 180 = π
Trigonometric Ratios and Identities
2. Pythagorean identity
• sin2 x + cos2 x = 1, sin2 x = 1–cos2 x and cos2 x = 1–sin2 x
• 1 + tan2 x = sec2 x, sec2 x–1 = tan2 x and sec2 x– tan2x = 1
• 1 + cot2 x = csc2 x, csc2 x–1 = cot2 x and csc2 x– cot2 x = 1
3. sec x + tan x, sec x – tan x are reciprocals to each other.
4. csc x – cot x, csc x + cot x are reciprocals to each other.
5. Trigonometric ratios of (– x)
• sin(–x) = –sin(x), cos (–x) = cos(x)
• csc(–x) = –csc(x), sec(–x) = sec(x)
• tan(–x) = –tan(x), cot(–x) = – cot(x)
6. Even trigonometric functions are cos x , sec x.
7. Odd trigonometric functions are sin x, tan x, csc x, cot x.
8. If a cos q + b sin q = c, then 222 sincos ababc θθ−=±+−
• If a cos q – b sin q = c ,then 222 sincos ababc θθ+=±+−
• 4422 2 sincos12sincos 1 1sin2 2 +=− =− θθθθ θ
• 6622 2 sincos13sincos 3 1sin2 4 +=− =− θθθθ θ
• sin2 q + cos4 q = cos2 q + sin4 q = 1–sin2 q cos2 q
9. If P n = cosn q + sinnq, then 22 2 2 sincos nn n PP P +θθ =
Compound Angles
For any A, B ∈ R,
10. () Sinsincoscossin ABABAB +=+
11 () Sinsincoscossin ABABAB −=−
12. () coscoscossinsin ABABAB +=−
13. () coscoscossinsin ABABAB −=+
14. ()()
22 22 sinsin sinsin coscos AB ABAB BA +⋅−=
15. ()() 22 22 cossin coscos cossin AB ABAB BA +⋅−=
16. Suppose that A, B, A + B are not the multiples of 2 π ; () tantan tan 1tantan AB AB AB + +=
17. Suppose that A, B, A – B are not the multiples of 2 π ; () tantan tan 1tantan AB AB AB −= +
18. Suppose that A, B, A + B are not the multiples of π; ()+= + cotcot1 cot cotcot AB AB BA
19. Suppose that A, B, A – B are not the multiples of π; ()+ −= cotcot1 cot cotcot AB AB BA
20. ()() 22 22 tantan tantan 1tantan AB ABAB AB +−=
21. ()() 22 22 cotcot1 cotcot cotcot BA ABAB BA +−=
22. For any A, B, C ∈ R,
• sin(A + B + C) = ∑ sinA cosB cosC –sinA sinB sinC
• cos( A + B + C ) = cos A cos B cos C –∑ sinA sinB cosC
23. For suitable angles of A, B, C, () tantan tan 1tantan AA ABC AB ∑−∏ ++= −∑
24. For suitable angles of A, B, C, () cotcot cot 1cotcot AA ABC AB ∑−∏ ++= −∑
25. Trigonometric ratios of 15°, 75°:
26. Trigonometric ratios of
q 21 21 + cot q 21 + 21
27. ()() sinsin120sin1200 θθθ ++°+−°=
28. ()() sinsin240sin2400 θθθ ++°+−°=
29. ()() coscos120cos1200 θθθ ++°+−°=
30. ()() coscos240cos2400 θθθ ++°+−°=
31. If A + B = 45° or A + B = 225° then
• (1 + tan A)(1 + tan B) = 2
• (1 – cot A)(1 – cot B) = 2
• (1 + cot A)(1 + cot B) = 2 cotA cotB
• 1 tan2221 2 °=−
32. If A + B = 135° or A + B = 315°, then
• (1 – tan A)(1 – tan B) = 2
• (1 + cot A)(1 + cot B) = 2
• (1 + tan A)(1 + tan B) = 2 tanA tanB
• 1 tan6721 2 °=+
33. If A + B + C is a multiples of π then
• tanA + tanB + tanC = tanA tanB tanC
• cotA cotB + cotB cotC + cotC cotA = 1
34. If A + B + C is multiples of 2 π then
• tanA tanB + tanB tanC + tanC tanA = 1
• cotA + cotB + cotC = cotA cotB cotC
Multiple and Submultiple Angles
35. sin(2A) = 2 sinA cosA
36. () 2 2tan tan2 1tan A A A = Here A, 2A are not odd multiples of 2 π
37. () 2 cot1 2cot cot2 1 cottan 2 A AA AA
=
. Here A, 2A are neither integral multiples of nor odd multiples of 2 π
38.
sin2sincos 22 AA A
39. 22 2 2 cossin 22 cos12sin 2 2cos1 2 AA A A A
=−
40. 2 2cos1cos 2 A A
41. 22 coscossin 22 AA A =−
42. 2 2sin1cos 2 A A
43. 2 2tan 2 tan 1tan 2 A A A = Here , 2 A A are not odd multiples of 2 π
44. 2 cot1 2 cot 2cot 2 A A A = Here , 2 A A are not odd multiples of 2 π
45. 2cotcottan 22 AA A =−
46. 2 2tan sin2 1tan A A A = +
47. 2 2 1tan cos2 1tan A A A = +
48. 2 2tan tan2 1tan A A A =
Here 2A is not odd multiples of 2 π
49. 2 2tan 2 sin 1tan 2 A A A
=
+
Here A is not an odd multiple of π.
50. 2 2 1tan 2 cos 1tan 2 A A A
= +
Here A is not an odd multiple of π.
51. 2 2tan 2 tan 1tan 2 A A A
=
Here A/2, A is not an odd multiple of , 2 π π .
52. sin 3A = 3 sinA – 4 sin3A
53. cos 3A = 4 cos3A – 3 cosA
54. 3 2 3tantan tan3 13tan AA A A = 3 A, A are not odd multiples of 2 π
55. 3 2 3cotcot cot3 13cot AA A A = 3A, A are not odd multiples of π
Here A is not an integral multiple of π and 2 A is neither an integral multiple of π nor an odd multiple of 2 π
56. For all A ∈ R, • 1cos2 sin 2 A A =±
• 1cos2 cos 2 A A + =±
57. If a1, a2, .......a n are in arithmetic progression with common difference d then
• sin a1 + sin a2 + …+ sin a n = 1 sinsin 22 sin 2 n aand d +
• cos a1 + cos a2 + …+ cos a n = 1 cossin 22 sin 2 n aand d +
58. When q is not multiple of π and not multiple of 2 π :
• ()() 1 coscoscoscos3 4 θθαθαθ +−=
• tan q tan( q + a ) tan ( a-q ) = tan 3 q
When q , 3 q are not odd multiples of 2 π
• tan q + tan(60° + q ) + tan(120° + q ) = 3tan 3 q
• tan q + tan(120° + q ) + tan(240° + q ) = 3tan 3 q
• tan q + tan(240° + q ) + tan(300° + q ) = 3tan 3 q
• tan q + tan(60° + q ) + tan(300° + q ) = 3tan 3 q
When q is not a multiples of π:
• cos q cos2 q cos4 q .....cos(2n–1q ) ()() 1 sin2 coscos2cos4...cos2 2sin n n n θ θθθθ θ =
• () ()()tan1sec1sec2...1sec2 2 tan2 n n θθθθ θ +++ = (1+ sec q )(1+ sec2 q ).... (1+ sec2nq ) = tan2nq
• () () () () 1 2cos12cos212cos41.... 2cos21 2cos21 2cos1 n n θθθ θθ θ + −= +
59. For any q∈ R,
• ()() 33 3 sinsin60sin60 3 sin3 4 θθθ θ +°−−°+ =−
• ()() 33 3 coscos60cos60 3 cos3 4 −°−−°+ = θθθ θ
60. If a = 60°, 120°, 240°, 300° then
• ()()222 3 sinsinsin 2 θαθαθ +−++=
• ()()222 3 coscoscos 2 θαθαθ +−++=
• ()() 1 sinsinsinsin3 4 θαθαθθ +−=
• If 21 n θπ = + then cos q cos 2 q cos 4 q() 1 1 cos2 2 n n θ=
• If 21 n θπ = then cos q cos 2q cos 4q.... () 1 1 cos2 2 n n θ=−
61. For all ()21;, 2 ARnnZ π ∈−+∈ 1cos2 tan 1cos2 A A A =± +
62. For all {} ; ARnnZ π ∈−∈ , 1cos2 cot 1cos2 A A A + =±
63. For all ()21; 2 ARnnZ π ∈−+∈ , 2 sec 1cos2 A A =± +
64. 1cos sin 22 AA =± For all A ∈ R
65. 1cos cos 22 AA + =± For all A ∈ R
66. 1cos tan 21cos AA A =± + For all ()21; 2 ARnnZ π ∈−+∈
67. Trigonometric ratios of some special angles:
• 1 cot76432 2 °=+++ • 1 tan76432 2 °=−−+
69. ()()2cos1si s n n 2 1i A A A =±±− ++
70. ()()2sin1si s n n 2 1i A A A =±±− +−
71. 11tan2 tan 2tan AA A −±+ =
72. 1sin 3 cossin 22424 37 cossin 22424 A AAA AAA ππ ππ +
−−<<
73. 1sin 3 cossin 22424 5 cossin 22424 A AAA AAA ππ ππ −−<<
−+<<
74. If 3 , 244 A ππ ∈ , then C + S > 0, C – S < 0
1sin1sin2sin 2 A AA ++−= and 1sin1sin2cos 2 A AA +−−=
1 67 2 o 21 22 + 21 22
68. Trigonometric ratios of some more special angles:
• 3555 sin9 4 °=+−−
• 3555 cos9 4 °=++−
75. If 57 , 244 A ππ ∈ , then C + S < 0, C – S > 0
1sin1sin2cos 2 A AA ++−=−
1sin1sin2sin 2 A AA +−−=−
76. Let cos,sin 22 AA CS==
• If , 244 A ππ ∈− , then C + S > 0,
C – S > 0
• If 3 , 244 A ππ ∈ , then C + S > 0,
C – S > 0
• If 35 , 244 A ππ ∈ then C+S < 0,
C–S < 0
• If 57 , 244 A ππ ∈ then C+S < 0,
C–S > 0
Transformations
For all C, D ∈ R,
CDCD CD +−
77. sinsin2sincos 22
CDCD CD +−
78. sinsin2cossin 22
CDCD CD +−
79. coscos2coscos 22
CDCD CD +− −=−
80. coscos2sinsin 22
81. 2sinA cosB = sin(A + B) + sin (A – B)
82. 2cosA sinB = sin(A + B) – sin (A – B)
83. 2cosA cosB = cos(A + B) + cos (A – B)
84. –2sinA sinB = cos(A + B) – cos (A – B)
85. In a triangle ABC,
• s in2 A + sin2 B + sin2 C = 4 sin A sin B sinC
• cos2 A + cos2 B +cos2 C = –1– 4 cos A cosB cosC
86. If cos x + cos y = a and sinx + siny = b, then
tan 2 xyb a + =
87. If cos x – cos y = a and sin x – sin y = b, then tan 2 xya b +
Periodicity and Extreme Values
88. A function f: A → B is said to be periodic function if there exists a real number p such that f ( x + p ) = f ( x ), ∀ x ∈ A . The smallest such value of p is called period of f or fundamental period of f.
90. If the period of a function f(x) is p, then f ( x + np ) = f ( x ) for all elements of x in the domain of f(x), where n is an integer.
92. If a function f(x) is periodic with period p, then period of function f(ax + b) + c is p a
94. The fundamental period of sin x , cos x , sec x, csc x is 2π.
95. The fundamental period of tan x, cot x is π.
96. The period of sinnx, cosnx, secnx, cscnx is π, when n is even.
97. The period of sinnx, cosnx, secnx, cscnx is 2π, when n is odd.
98. Period of a rational function of functions: The period of ()()() ()() 1122 3344 CfxCfx gxCfxCfx + = + is LCM of periods of f1, f2, f3, f4.
99. The extremum values of k sin x, k cos x are k, –k.
100. The extremum values of a cosx + b sin x are 2222 , abab +−+ .
101. The extremum values of a sin2 x + b sin x cos x + c cos2 x are ()2 2 22 acbac ++− ±
102. a2 sec2 x + b2 cosec2 x ≥ (a + b)2 occurs at 1 tan b x a =
JEE MAIN LEVEL
Level – I
Trigonometric Ratios and Identities
Single Option Correct MCQs
1. If cot q = 3 4 and q is not in the second quadrant, then 5sin q +10cos q + 9sec q +16cosec q – 4cot q= (1) 0 (2) –1
(3) 1 (4) 2
2. If tan q= 4 3 , then sin q is
(1) 44 but not 55 (2) 44 or 55 (3) 44 but not 55 (4) 33 or 55
3. If 1 sincos 5 θθ+= and 0, θπ≤< then tanq is
(1) –4/3 (2) –3/4 (3) 3/4 (4) 4/3
4. If 5 sinx + 4 cosx = 3, then 4 sinx – 5 cosx = (1) 4 (2) 42 (3) 32 (4) 2
5. The value of 2 sec1 cot 1sin α α α + 2 sin1 sec 1sec + + α α α is (1) 0 (2) 1 (3) 2 (4) –2
6. (1 + tan a . tan b )2 + (tan a – tan b )2 = (1) tan2a . tan2b (2) sec2a . sec2b (3) tan2a . cot2b (4) sec2a cos 2b
7. If sinx + siny +sinz +sinw = –4, then the value of sin400x + sin 300y + sin200z + sin100w is
(1) sin400x . sin 300y . sin200z + sin100w
(2) sinx . siny . sinz . sinw (3) 4 (4) 3
8. If sinx + sin2x = 1, then cos2x + cos4x = (1) 0 (2) 1 (3) 2 (4) –1
9. If cosx + cos2x = 1, then sin8x + 2sin6x + sin4x is (1) 0 (2) 1 (3) 2 (4) –1
10. Which of the following is correct? (1) sin1° > sin1 (2) sin1° < sin1
(3) sin1° = sin1 (4) sin1° = 180 π sin1
11. If y = cos2q + sec2q , , nZ≠∈θπ , then (1) y = 0 (2) y ≤ 2 (3) y ≥ –2 (4) y > 2
12. The value of cosec(75° + q ) – sec(15° – q ) – tan(55° + q ) + cot(35° – q )is
(1) –1 (2) 0 (3) 1 (4) 3 2
13. The value of x, in which sincosxx is defined in [0, 2 π ], is (1) 0, 4 π
(3) 5 , 44 ππ
(2) 57 , 44
(4) 0, 2 π
14. Which of the following is greatest? (1) cosec 1 (2) cosec 2 (3) cosec 4 (4) cosec (–6)
15. If K is a positive integer, then sin( a + π ) + sin( a + 2 π ) + sin( a + 3 π ) +....+ sin( a + 2K π ) = (1) 0 (2) 1 (3) sin a (4) – sin a
16. 42 4 2 1 sin2sin1 cos Cos ec +−+=
θθθ θ (1) 1 (2) 0 (3) 1 2 (4) –1
17. If 3 cos 5 = θ and q is not in the first quadrant, then ()() ()() 5tan4cos 5sec24cot2 +++ = −−+ πθπθ
(1) 4 5 (2) 4 5 (3) 5 4 (4) 5 4
18. The value of 2 1sin140sec140 −°° is (1) 1 (2) –1 (3) 0 (4) –2
19. If ABCD is a quadrilateral, then 22 seccot 44 ABCD ++
(1) 0 (2) 1 (3) –1 (4) 2
20. cos210° + cos30° + cos20° + cos200° =____. (1) 1 (2) 2 (3) –1 (4) 0
21. () () 2 33 tantancos 22 . 3 sin cos 2 xxx x x
sin2(2 π – x) = (1) sin2x (2) cos2x (3) – cos2x (4) –sin2x
22. If cot( a + b ) = 0, then sin( a + 2 b ) = (1) –sin a (2) sin a (3) ±cos b (4) both (2) and (3)
23. If ()() tan 3 , tan – 1 αβαβ +== , then tan6 b= (1) –1 (2) 0 (3) 1 (4) 2
24. If sin( a + b ) = 1 and sin( a – b ) = 1 2 , then tan ( a + 2 b ) tan(2 a + b ) = (1) 1 (2) –1 (3) 0 (4) 1 2
25. If a, and b are complementary angles, sina = 3 5 then sin a cos b – cos a sin b =_____.
(1) 7 25 (2) 7 25 (3) 25 7 (4) 25 7
26. If a, and b are supplementary angles, then cos2a+ sin2b =______.
(1) 1 (2) –1 (3) 2 (4) 0
27. 379 tantantantan 10101010 +++= ππππ _______.
(1) 0 (2) 1 (3) –1 (4) 2
28. 0000 sin24sin32sin204sin212 +++= _____.
(1) –1 (2) 1 (3) 0 (4) 2
29. cos 5° + cos 24° + cos 175° + cos 204° + cos 300° =______. (1) 1 2 (2) 1 (3) 2 (4) 0
30. If 32 3 sincos tantantan cos xx axbxcxd x + =+++ then a + b + c + d = (1) 4 (2) –4 (3) 1 (4) –1
31. If q is acute and (1 – a2)sinq = (1 + a2)cosq, then sin q =
(1) () 2 4 1 21 a a + (2) () 4 2 21 1 a a + (3) () 2 4 1 21 a a + + (4)
32. If 8 cosx + 15sinx = 15 and cosx ≠ 0, then 8 sinx – 15 cosx =
(1) 8 (2) –8 (3) 15 (4) –15
33. If sinq + cosq = 3 cosq, then 2cosq – sinq =
(1) 3 sin q (2) – 3 sin q
(3) – 3 cos q (4) 3 sin q
34. 2 2 If ; tan then n nn n xasec ybxy ab θ θ =
(1) 0 (2) –1 (3) 1 (4) 2
35. If a = x sec q + y tan q , b = x tan q + y sec q, then x2 – y2 = (1) a2 – b2 (2) a2 + b2 (3) b2 – a2 (4) a2b2
36. If sin q + cos q =p and tan q + cot q =q, then q(p2 –1) = (1) 1 2 (2) 2 (3) 1 (4) 3
37. If x = a(cosecq + cotq), y = b(cosecq – cotq) then (1) x + y = ab (2) x – y = ab (3) xy = ab (4) xy = 0
38. 2222 If coscoscos ; coscossin ; sincos, sin then xr yr zrr xyz αβγ αβγ αβµβ µ = = == +++= (1) r (2) 2r (3) r2 (4) 4r2
Numerical Value Questions
39. If 3 sec a – 5 tan a = k and 6 sec a + k tan a = 5, then k2 = _______.
40. If 3333 cossincossin cossincossin AAAAk AAAA +− += +− , then k =____
41. If tanA + cotA = 4, then tan4A + cot4A =____
42. (sin a + cosec a ) 2 + (sec a + cos a ) 2 = k + tan2a +cot2a, then k = _____.
43. The number of points of intersection of the graphs y = sinx and y = cosx in [0, 2 π ] is ____.
44. ()() 22 sin51sin39xx −++=
45. The value of cosec (–1410)° is ______.
46. If sec x + sec 2 x = 1, then the value of tan 8 x – tan 4 x –2tan 2 x + 1 will be equal to_____.
47. If 4242 secsec10tantan θθθθ +=++ ,then 5sin2 θ = _______.
48. For 0, 2 A π << the value of 1 22 2 12 log 12cossec2 AA + ++ = _______.
49. sin7sin3 If then 21sin24sin14 πα−α α== α+α ______.
Compound Angles
Single Option Correct MCQs
50. 22 cossin 66 = π +θ−−θ π (1) 1 cos2 2 θ (2) 1 cos2 2 −θ (3) 0 (4) 1
51. If 41 tanA,tanB 37 == , then A – B = (1) π 4 (2) π 2 (3) π (4) π 2 3
52. If == 171 tan, tan 1835 AB , then cos(A + B) = (1) 1 (2) 2 (3) –1 (4) 1 2
53.
cosA.cosBsinA.sin 4444 B
(1) cos(A + B) (2) cos(A – B) (3) sin(A + B) (4) sin(B –A)
54. In ∆le ABC, cosA + cos(B – C) = (1) 2sinBsinC (2) 2cosBsinC (3) 2sinBcosC (4) 2cosBcosC
55. If sin( q + a ) = cos( q + a ), then tan q = (1) 1tan 1tan +α −α (2) 1sin 1sin +α −α (3) 1tan 1tan −α +α (4) 1cos 1cos +α −α
56. If += π AB 2 , then tanB + 2tan(A – B) = (1) cosA (2) sinA (3) tanA (4) cotA
57. tan20° + 2tan50° = (1) tan30° (2) cot20° (3) tan20° (4) cot40°
58. If tan40° + 2tan10° = cot x, then x = (1) 75° (2) 85° (3) 30° (4) 40°
59. If () 74 tan,tan 243 ABA−== , where A, and B are acute then A + B = (1) π 5 (2) π 4 (3) π 3 (4) π 2
60. If ()−= 3 cosAB 5 and tanAtanB = 2, then which one of the following is true ? (1) ()+= 1 sin AB 5 (2) ()+= 1 sin AB 5 (3) () 1 cos A+B 5 = (4) () 1 cos A+B 5 =
61. ()() ()() +α+−α = +α+−α tan45tan45 cot45cot45 (1) 1 (2) 2 (3) 3 (4) 4
62. If 0 2 <θ<π and 2sin3cos10sin10θ=+ , then q = (1) 40° (2) 50° (3) 70° (4) 80°
63. += + cos13sin131 cos13 sin13cot148 oo oo (1) 1 (2) –1 (3) 0 (4) 1 2
64. cos40° + cos80° + cos160° + cos240° = (1) 0 (2) 1 (3) 1/2 (4) –1/2
65. tan25°.tan31° + tan31°.tan34° + tan34° tan25° = (1) 1 (2) 2 (3) 3 (4) 4
66. If tanA = 1, tanB = 2, tanC = 3, then A + B + C = (1) π ∈ ,nz 2 n (2) n π , n ∈ z (3) π ∈ ,nz 4 n (4) π ∈ 2 ,nz 3 n
67. ()() θ+θ = π θ+ 3sincos sin 6 (1) –2 (2) 1 (3) 2 (4) –1
68. tan5x – tan3x – tan2x = (1) tan5x.tan3x.tan2x (2) sin5x.sin3x.sin2x (3) cos5x.cos3x.cos2x (4) cot5x.cot3x.cot2x
69. If α=β= ++ 1 tanandtan 121 m mm , then the least possible value of α + β is (1) π 4 (2) π 3 (3) π 6 (4) π 15
70. The value of tan32°.tan23° + tan23°.tan35° + tan35°.tan32° is
(1) less than 1 (2) greater than 1 (3) equal to 1 (4) equal to 3
Numerical Value Questions
71. If () cotA3 cotB2andcosAB 5 =+= then 5sinAsinB = _______.
72. tan20° + tan25° + tan20° tan25° =
73. If tan22tan383 tan22tan38 k +−=− then k2 = _____.
74. The value of () () () () ++00 00 1tan221tan23 1cot331cot12 is
75. = 00 0 tan80tan10 tan70
76. If A = 35°, B = 15°, and C = 40°, then tanAtanB + tanBtanC + tanCtanA= _____.
77. In an acute angled triangle, cot B cot C + cotAcotC + cotAcotB =__________
78. If cos(x – y) = 3 cos(x + y), then cotx coty = ________.
Multiple and Submultiple Angles
Single Option Correct MCQs
79. 44 442 cos2cossin cossin2sin2 ααα ααα + −= (1) 0 (2) 1 (3) 1/2 (4) 2
80. θ+θ = +θ+θ sinsin2 1coscos2 (1) θ tan 2 (2) θ t 2 co (3) tan q (4) cot q
81. 3cosec20sec20−= (1) 2 (2) 3 (3) 1 (4) 4
83. cos2( q – 45°) + cos2( q + 15°) –cos2( q – 15°) = (1) 1 2 (2) 1 3 (3) 1 2 (4) 1 3
84. If x cosq = ycos(q + 120°) = z cos(q + 240°) then xy + yz + zx = (1) –1 (2) 2 (3) 0 (4) 3 4
85. If sin2q = cos3q is an acute angle, then sinq equals (1) 51 4 (2)
51 4 (3) + 51 4 (4) 51 4
86. sin12° sin24° sin48° sin84° = (1) cos20° cos40° cos60° cos80° (2) sin20° sin40° sin60° sin80° (3) 3 15 (4) 1 2
87. cos276° + cos216° – cos76°cos16° = (1) 1 2 (2) 0 (3) 1 4 (4) 3 4
88. If sin( π cos q ) = cos( π sin q ) then sin2 q = (1) 3 4 ± (2) 1 3 ± (3) 2 3 ± (4) 1 4 ±
82. 2cosec20sec20 = (1) 2 (2) 2sin20°cosec40° (3) 4 (4) 4sin45°cosec40°
89. A ∈ Q3, 2 75 tan1816sin32sinsin 3222 AAA A =⇒−−= 2 75 tan1816sin32sinsin 3222 AAA A =⇒−−= (1) –6 (2) 11 (3) 5 (4) 10
90. If 2sec 2 a = tan b + cot b , then one of the values of α + β is (1) 4 π (2) 3 π (3) 2 π (4) π
91. cos66° + sin84° = (1) 153 4 (2) 153 4 (3) 153 4 + (4) 153 4 +
Numerical Value Questions
92. 300 cosec20sec20−= _____.
93. πππππ = m cos cos coscos 2 481632n cosec then m + n = ____.
94. 3 3If2cos20then11 xx xx +=°+= ____.
95. ()() ()() 00 00 tan45 tan45 cot 45 cot45 −α−+α = +α−−α
96. If 4sin(60° + q )sin(60° – q ) – 1 = kcos2 q , then k = _____.
97. If ππππ +++= 44445711 cos cos cos cos 12121212 k then 4k =
98. 2(cos36° – cos72°) =______.
99. tan9° – tan27° – tan63° + tan81° =
100. If +=λ 11 , cos290 3sin250 then the value of 9 λ 4 + 81 λ 2 + 97 must be ______
101. + = 0000 20 cos208sin10sin50sin70 sin80 ______.
Transformations
Single Option Correct MCQs
103. sin3sin5sin7sin9 cos3cos5cos7cos9 θ+θ+θ+θ = θ+θ+θ+θ
(1) tan4 q (2) tan6 q
(3) tan2 q (4) cot6 q
104. () () + == tan If sin2sin2,then tan xy xny xy
(1) 1 1 n n + (2) 1 1 n n + (3) 1 1 n n + (4) 1 1 n n +
105. If cosx + cosy + cosa = 0, sinx + siny + sina = 0 then cot 2 xy +
(1) sin a (2) cos a
(3) tan a (4) cot a
106. cos20° + cos30° + cos40° =
(1) 1 – 2sin10°sin15°sin20°
(2) cos20°cos30°cos40°
(3) 4 cos10°cos15°cos20°
(4) 4cos25°cos30°cos35°
107. 1 + cos10° + cos20° + cos30° = (1) 4cos5°cos10°cos15°
(2) 4cos10°cos20°cos30°
(3) 4sin5°sin10°sin15°
(4) 4sin10°sin20°sin30°
108. If sin10°sin50°sin60°sin70° = m and tan20°tan40°tan60°tan80° = n, then n m =
(1) 33 16 (2) 163
(3) 16 3 (4) 83
102. sin5sin3 cos52cos4cos3 α−α = α+α+α (1) t 2 co α (2) cot a (3) tan 2 α (4) none
109. The value of sin 310° + sin350°– sin370° is equal to (1) 2 3 (2) 3 4 (3) 3 4 (4) 3 8
110. 357 sinsinsinsin 14141414 ππππ ⋅⋅⋅⋅ 91113 sinsinsin 141414 πππ ⋅⋅=
(1) 1/64 (2) 3/64 (3) 5/64 (4) 7/64
111. If x , y , z are in A.P, then sinsin coscos xz zx is equal to (1) tan y (2) cot y (3) sin y (4) cos y
112. If cos q 1 = 2cos q 2, then ()() tantan1212 22 θ+θθ−θ is equal to (1) 1/3 (2) –1/3 (3) 1 (4) –1
113. The value of 2015 2015 coscossinsin sinsincoscos ABAB ABAB ++ += (1) 0 (2) 2015 cot 2 AB +
(3) 2015 cot 2 AB (4) 2tan2015 2 AB +
116. 42 coscos,coscos 57xyxy +=−= , then the value of 14.tan5.cot 22 xyxy −+ +
is (1) 0 (2) 1 (3)–1 (4) 2
117. The value of the expression 00 0 14.sin10.sin70 2.sin10 (1) 1/2 (2) 1 (3) 2 (4) 1/3
Numerical Value Questions
118. sin10° + sin20° + sin40° + sin50° – sin70° – sin80° =_________.
119. If x + y + z = 180°, then sinsinsin coscoscos 222 xyz y xz ++ = _______.
Periodicity and Extremum Values
Single Option Correct MCQs
120. The period of x – [x] is_____ (where [x] represents the integral part of x) (1) 1 2 (2) 1 (3) 1 3 (4) 2
121. The period of 3x – [3x] is (where [.] denotes greatest integer function ≤ x) (1) 1 2 (2) 1 (3) 2 (4) 1 3
114. If ()()() coscos2cos3 cos xxxx abcd +θ+θ+θ === )()() coscos2cos3 xxx abcd +θ+θ+θ === , then ac bd + = + (1) b c (2) a d (3) d a (4) c d
115. sinA + sin3A + sin5A + sin7A = (1) 4sinAcos2Acos4A (2) 4sinAcos2Acos3A (3) 4cosAsin2Asin4A (4) 4cosAcos2Asin4A
122. The period of 22 sin3cos xx ab ππ + , when a = 12, b = 9, is (1) 18 (2) 36 (3) 108 (4) 54
123. The period of tan (x + 8x + 27x + …+n3x) is (1) 22 8 (1)nn π + (2) 22 4 (1)nn π + (3) 22 2 (1)nn π + (4) () 4 1 nn π +
124. The period of cottan 44 1tantan 2 xx x x + +− is (1) 2 π (2) π (3) 4 π (4) 2 π
125. The period of 3sin5x + cos3x is (1) π (2) 2 π (3) 2 π (4) 3
126. Period of sin cos 11 33
127. The range of 2222 cos cos 33 xx ππ
128. If 3sincosxx + is maximum, then x = (1) 4 π (2) 3 π (3) 2 5 π (4)
16cos5x – 20cos3x + 5cos
130. cos2(60°– x) + cos2(60° + x) ∈ (1) 11 , 22
(3) 13 , 22
Numerical Value Questions
131. Period of cos2cosec5tan 463 xxx πππ −+ is _____.
132. Period of 32 tan2sec5sin 435 xxx
is _______.
133. The period of () () sin2 sin2 xa xb π π + + is______.
134. The period of function f(x) = {x} + tan2πx + |sin3πx| is ______.
135. The maximum value of 66 1 , sincos y xx = + , is _________.
136. Minimum value of 9sec2q + 4cosec2q + 7 is _______.
137. sin2x + 4sinx+5 ∈ [k,5k] ⇒ k=
Level – II
Trigonometric Ratios and Identities
Single Option Correct MCQs
1. If tan2a = 1 – p 2, then sec a + tan3a cosec a =
2. If sincostan , xxxk abc === then 1 1 bcak ckbk ++ + is equal to (1) 1 ka a
+
+
(2) 11 a ka
(3) 2 1 k (4) a k
3. If 1 + sin q = 9 cos q and 0° < q <90°, then 5(1 – cos q ) = (1) sin q (2) 2 sin q (3) 4 sin q (4) tan q
4. cos q + cos( π + q ) + cos(2 π + q ) +....+ cos(20 π + q ) = (1) 0 (2) cos q (3) sin q (4) –cos q
5. παα α αα == + sin3sin23 If , then 19sin4sin16 (1) 1 (2) –1 (3) 0 (4) 1 2
6. If 24 cos cos 33xyz
ππ , then xy +yz +zx =_____.
(1) 0 (2) 1 (3) –1 (4) 1 2
7. If () sin246 sin sin log2 8 xxx e +++…∞ = and 0 2 x << π , then cosx cosx+sinx = (1) 31 2 + (2) 31 2 (3) 2
8. If sinq, cosq, and tanq are in GP, then cos9q + cos6q + 3 cos5q = (1) 1 (2) –1 (3) 0 (4) 2
9. If π < α < 2 π , then 22 1 Sincotcosααα = (1) sinα (2) –sinα (3) 1 sin α (4) 1 10. 3 3 32cos12sin 12sin 32cos AA AA
(1) 1 (2) 3 (3) 0 (4) –1
11. If m = sin q + cos q ; and n = sec q + cosec q, then
(1) n(m2 + 1) = 2m
(2) n(m2 – 1) = 2m
(3) 2n(m2 + 1) = m
(4) 2n(m2 – 1) = m
12. The equation 22 cosec 2 xy xy θ+ = is () 0,0xy≠≠
(1) possible for x = y (2) possible for all real values of x and y (3) impossible for all real values x and y (4) can’t be determined
13. If sin q + sin2q = 1 and acos12q + bcos10q + ccos8q + dcos6q –1 = 0, then bc ad + = +
(1) 1 (2) 2 (3) –2 (4) 3
14. If sinq + cosq = 3 cosq, then 2cosq – sinq = (1) 3 sin q
(2) – 3 sin q
(3) – 3 cos q (4) 3 sin q
Numerical Value Questions
15. If sinA, cosA, and tanA are in GP, then cot6A – cot2A =_______.
16. If 2 πθπ << , () 1sin1sin , 1sin1sin f −+ =+ +− θθθ θθ then 2 3 f π = _______.
17. The minimum value of the function () 22 22 sincos 1cos1sin tancot sec1cosec1 xx fx xx xx xx =+ ++ (whenever it is defined) is ______
18. If sinx + sin2x + sin3x = 1, then cos6x – 4cos4x + 8cos2x = ______.
19. cosec2Acot2A – sec2Atan2A – (cot2A – tan2A) (sec2Acosec2A – 1) =_____
20. 42 3(sincos)6(sincos) xxxx −+++ () 66 4sincos is equal to xx + ______.
21. If sin b is the GM between sin a and cos a , then |(cos a- sin a )2 – 2cos2b|=_____.
22. For 0, 2 A π << the value of 1 22 2 12 log 12cossec2 AA + ++ = _______.
23. The minimum value of 1 2cos 2tan sin θθ θ ++ in 0, 2 π is____
24. If sec6q – tan6q = a sec4q + bsec2q + c then a + b + c = _______.
25. If cosx + cos2x = 1, then sin12x + 3 sin10x + 3 sin8x +sin6x + 1= _______.
26. sin10sin20sin30sin360 +++……+=
27. 2222 248 sinsinsinsin 18181818 ππππ +++ sinsin2275 1818 ππ ++=
28. If ()()() ()()() 000 00 sin660tan1050sec420 cos225cosec315cos510 k = then 3k = _______.
Compound Angles
Single Option Correct MCQs
29. The value of tan40° + tan11° + tan20° –tan56° + tan56° tan11° + = 00 3 tan40 tan20
(1) 31 (2) + 31 (3) 1 (4) 0
30. If B, A + B are acute angles, () 125sinB,sinB 1313 A +== , then sinA = (1) 119 169 (2) 119 169 (3) 169 119 (4) 169 119
31. If ππ +θ+−θ= tan tan a 44 then ππ
tan33 tan 44
(1) 0 (2) a (3) 3a (4) a3 – 3a
32. cosx,cos,cosx 33 x −+ π π are in HP, then cosx = (1) 3 2 (2) 1 (3) 3 2 (4) 3 2
33. In a ∆ le ABC if == A5B20 tan, tan 26237 , then = C tan 2 (1) 5 2 (2) 2 5 (3) 307 122 (4) 7 4
34. If () 11 0A,B,cosB 4 61 A + π <<= and ()−= 24 sinAB, 25 then sin2A + sin2B =
(1) 684 1525 (2) 156 1525
(3) 168 305 (4) 168 1525
35. tan(A – B) + tan(B – C) + tan(C – A) =
(1) tanA tanB tanC
(2) cotA cotB cotC
(3) tan(A – B)tan(B – C)tan(C – A) (4) 0
36. In any triangle ABC, sin2A – sin2B +sin2C is always equal to
(1) 2sinAsinBcosC (2) 2sinAcosBsinC
(3) 2sinAcosBcosC (4) 2sinAsinBsinC
37. The expression cos2( a + b ) + cos2( a – b ) – cos2 a cos2 b is (1) –1
(2) 2
(3) independent of a and b (4) dependent on a and b
38. The value of cos 2 76° + cos 216°– cos 76°. cos16° =_____.
(1) 3 4 (2) 1 4 (3) 0 (4) 1 2
39. In a triangle ABC, if tanA + tanB + tanC = 6 and tanAtanB = 2 then the triangle is (1) right angled (2) isosceles (3) acute angled (4) obtuse angled
40. If () sinAsinB3cosBcosA +=− then
Sin3A + Sin3B = (1) 0 (2) 2222 (3) 1 (4) –1
41. If A + B + C = 720°, then tanA + tan B + tanC = (1) tanA tanB tanC
(2) tan2A tan2B tan2C (3) tan2A + tan2B + tan2C (4) cotA cotB cotC
42. In ∆ABC, (cotB + cotC)(cotC + cotA) (cotA + cotB) =
(1) secA secB secC
(2) cosecA cosecB cosecC
(3) tanA tanB tanC
(4) 1
43. () cosABC cotA+cotB+cotC sinAsinBsinC ++ +=
(1) cotA cotB cotC
(2) tanA tanB tanC
(3) cosA cosB cosC
(4) sinA sinB sinC
44. If () () () () θ+θθ−θ += θ−θθ+θ 1234 1234 coscos 0, coscos then
cot q 1 cot q 2 cot q 3 cot q 4 = ____.
(1) 1 (2) –1 (3) 2 (4) 1/2
Numerical Value Questions
45. If 2tanA + cotA = tanB then cotA + 2 tan (A – B) =
46. If α=β=− 5 cot1,sec 3 when ππ π<α<<β<π 3 , 22 then the quadrant in which a + b lies is ____
47. In a triangle ABC, () ∑= cosBC sin B sinC ____.
48. If == tan tantan , 235 y xz x + y + z = π and ++=222 38 tantantanxyzK , then k = _____.
49. Let a,b,γ > 0 and 2 α+β+γ=π . If p = 7 + tan a tan b, q = 5 + tan b tan γ and r = 3 + tan γ tan a . Then the maximum value of () 1 3 pqr ++
50. If cot( q – a ), 3cot q , cot( q + a ) are in AP (where, π θ≠α≠π∈ , , ) 2 nkkI then θ α 2 2 2sin sin is equal to ______
51. Let πππ === 24 ,, 777 ABC and
cosA cosB cosC = –1/8 then | ∑ tanAtanB| is ______.
Multiple and Submultiple Angles
Single Option Correct MCQs
52. = sin sin 8 x x
(1) 8coscoscos 842 xxx
(2) 8cossinsin 842 xxx
(3) 8sinsinsin 842 xxx
(4) 8sinsincos 842 xxx
53. 24 tan2tan4cot 555 πππ ++=
(1) π 5 cot (2) π 2 5 cot (3) π 3 cot 5 (4) π 4 5 cot
54. If 5(tan2x – cos2x) = 2cos2x + 9, then the value of cos4x = (1) 7 9 (2) 3 5 (3) 1 3 (4) 2 9
55. The value of 4 sin27° (1) +−+ 5535
(2) +−− 5535 (3) −−+ 5535 (4) +−− 5535
56. If sina + sinb = a and cosa + cosb = b, then
(1)
α−β+=± 22 sin 24 ab
(2)
α−β+ = 22 cos 24 ab
(3)
α−β−−=± + 22 22 4 tan 2 ab ab
(4)
α−β++=± + 22 22 4 tan 2 ab ab
57. If ()−θ+θ θ= θ 1sin22 , 2cos2 fCos then value of 8f(11°). f(34°) is ______ (1) 4 (2) 1 (3) 3 (4) 2
58. If A = tan6°tan42°, B = cot66°cot78°, then (1) A = 2B (2) A = B (3) B = 2A (4) 3A= 2B
59. If = 0 51 sin18 4 , then sin 81° =
(1) +− + 511025 4242
(2) ++ + 511025 4242
(3) −+ + 511025 4242 (4) + 511025 4242
60. cos9° – sin9° = (1) 55 2 (2) + 55 4
(3) 1 55 2 (4) 55
61. If x 1 and x 2 are two distinct roots of the equation a cos x + b sin x = c , then + 12 tan 2 xx is equal to (1) a b (2) b a (3) c a (4) a c
Numerical Value Questions
62. If α=β=35 cos cos 513 and then 65 α−β
2 cos 2
63. If ⋅= 1 tantan 2 AB , then (5 – 3cos2A) (5 – 3cos2B) =
64. tan46° tan14° – tan74° tan14° + tan74° tan46° is equal to____.
65. If 7315 cot2cotcot 16816 K πππ ++= . Then 2 2 K = _____.
66. If x ∈ [–20°,–5°] and f(x) = tan(50° + x) + cos(50° + x ) + cot(50° + x ), then global maximum value of f(x) is k. Then the value of [k2] is _____(where [.] denotes greatest integer function).
67. If ππππ +++=4444 357 cos cos cos cos 888 8 k then 2k = ____
Transformations
Single Option Correct MCQs
68. If 0, 2 AB π << satisfy the equations 3 sin2A + 2sin2B = 1 and 3sin2A – 2sin2B = 0, then A + 2B = (1) 0 (2) 2 π (3) 6 π (4) 3 2 π
69. cos66cos415cos210 cos55cos310cos xxx xxx +++ = ++
(1) cosx (2) sinx (3) 2 sinx (4) 2 cosx
70. Ifcostantan ,then cos xAxAyB yBxy + == +
(1) tan 2
AB + (2) tan 2 AB
(3) t 2 AB co + (4) t 2 AB co
71. If coscos cos 1coscos θ=α−β −αβ , then
22 tantan 22 is
(1) tan 2 α (2) tan2 2 α
(3) t 2 co α (4) 2 t 2 co α
72. If A + B + C = 180°, then sin3A + sin3B + sin3C = (1) 333 4coscoscos 222 ABC (2) 333 4coscoscos 222 ABC (3) 333 14coscoscos 222 ABC
(4) 333 14sinsinsin 222 ABC
73. If 245730 coscoscoscoscoscos 151515151515 x ππππππ = then 1 8 x = (1) 4 (2) 1/4 (3) 8 (4) 4/3
74. If q 1, q 2, q 3, .... q n are in A.P. then 12 12 sinsinsin coscoscos n n θ+θ+…+θ = θ+θ+…+θ
(1) 0 (2) tan( q 1 + q n) (3) 1 tan 2 n θ+θ (4) 1 tan 2 n θ−θ
75. sin2sin2sin2 In , sinsinsin ABC ABC ABC ++ ∆= ++
(1) 4 sinA/2sinB/2sinC/2
(2) 4 cosA/2cosB/2cosC/2
(3) 8 sinA/2sinB/2sinC/2
(4) 1 + 4 sinA/2sinB/2sinC/2
76. If A + B + C = 180° then sin2sin2sin2 coscoscos1 ABC ABC ++ = ++−
(1) 4 cosA/2cosB/2cosC/2
(2) 4 sinA/2sinB/2sinC/2
(3) 8 cosA/2cosB/2cosC/2
(4) 1 + 4 sinA/2sinB/2sinC/2
Periodicity and Extremum Values
Single Option Correct MCQs
77. The period of sin(sinx) + sin(cosx) is (1) π (2) 2 π (3) 2 π (4) 4 π
78. Let () cos. fxpx = where p = [a] (integral part). If the period of f(x) is π then a ∈ (1) [4, 5] (2) [4, 5) (3) (4, 5] (4) (4, 5)
79. If A = sin2x + cos4x then ∀ x ∈ R (1) 13 1 11 A ≤≤ (2) 1 ≤ A ≤ 2 (3) 313 416 A ≤≤ (4) 3 1 4 A ≤≤
80. The maximum value of (cosα 1)(cos α 2)… (cosαn) under the restrictions 12 0,. 2 n π ααα ≤……≤ and cotα1. cotα2…cotα n = 1 is (1) /2 1 2n (2) 1 2n (3) 1 2n (4) 1
81. If A = sin 8θ + cos 14θ, then for all values of θ,
(1) 0 < A ≤ 1 (2) 1 < 2A ≤ 3
(3) A ≥ 1 (4) 1 0 2 A ≤≤
Graphs
Single Option Correct MCQs
82. The graph y = sin x is incresing in the interval is
(1) , 22
ππ (2) 3 , 22 ππ
(3) 3 , 22 ππ
(4) None
83. The asymtotes of the graph y = tanx is
(1) x = 2 π (2) x = 0
(3) y = –x (4) None
84. For a given integer K, in the interval 2,2 22 kkππ
the graph of sinx is
(1) increasing from –1 to 1
(2) decreasing from –1 to 0
(3) decreasing from 0 to 1
(4) None of these
Level – III
Single Option Correct MCQs
1. If secα and cosecα are the roots of x2 – px + q = 0, then (1) p2 = q(q–2) (2) p2 = q(q+2) (3) p2 + q2 = 2q (4) p2 – q2 = 2
2. If x = sin1, y = sin2, z = sin3, then (1) x < y < z (2) x > y > z
(3) x < z < x (4) z < x < y
3. Let () 1 1(cot) fx = + θ θ and () 0 0 89 1 Sf = =∑ θ θ .
Then the value of 28 S −= (1) 8 (2) 9 (3) 10 (4) 12
4. If u n = cosnq + sinnq and u n –u n – 2 = Ku n - 4, then K =
(1) 1 (2) sin2q
(3) – sin2q cos2q (4) cos2q
5. If asin3x + bcos3x = sinx cosx and asinx = bcosx, then a2 + b2 =
(1) 1/2 (2) 2 (3) 1 (4) 1/3
6. If a1cosq + b1sinq + c1= 0 and a2cosq + b2sinq + c2= 0, then (b1c2 – b2c1)2 + (c1a2 – c2a1)2 =
(1) (a1b2 – a2b1)2 (2) (a2b2 – a1b1)2
(3) (c1 – c2)2 (4) (b2c2 – b1c1)2
7. A quadratic equation whose roots are 00 11 tan22andcot22is 22
(1) 2 –22 1 0 xx +=
(2) 2 2–22 1 0 xx +=
(3) 2 22 1 0 xx+−=
(4) 2 –22 1 0 xx −=
8. cos a .sin( b – γ ) + cos b sin( γ – a ) + cos γ . sin( a – b ) = (1) 0 (2) 1 (3) –1 (4) 4 cos a cos b cos γ
9. () cosABC cotA+cotB+cotC sinAsinBsinC ++ +=
(1) cotA cotB cotC (2) tanA tanB tanC (3) cosA cosB cosC (4) sinA sinB sinC
10. If a + b + γ = 2 π , then
(1) αβγαβγ ++= tantantantantantan 222222
(2) αββγγα ++= tantantantantantan1 222222
(3) αβγαβγ ++=− tantantantantantan 222222
(4) αβ ∑= tantan1 22
11. If cos2A + cos2B + cos2C = 1, then ∆le ABC is
(1) equilateral (2) isosceles
(3) right angled (4) right angled isosceles
12. In ∆ le ABC if cos A cos B cos C = 1/3 then tanAtanB + tanBtanC + tanCtanA = (1) 1 (2) 4 (3) 3 (4) 1/3
13. () sinABC tanA+tanB+tanCcosAcosBcosC ++ =
(1) tanAtanBtanC (2) sinAsinBsinC (3) cosAcosBcosC
(4) tanAtanB + tanBtanC + tanCtanA
14. The value of 1sin21sin2 1sin21sin2 AA AA ++− +−− is, when |tanA| < 1 and |A| is acute. (1) cotA (2) sinA (3) cosA (4) secA
15. = π ∑= 10 3 1 cos 3 r r (1) 1 8 (2) 7 8 (3) 9 8 (4) 1 8
16. If cos a + cos b = a , sin a + sin b = b and a – b = 2 q , then θ = θ cos3 cos (1) a2 + b2 – 2 (2) a2 + b2 – 3 (3) 3 – a2 + b2 (4) (a2 + b2)/4
17. Iftan3sin3 ,then tansin AA a AA ==
(1) 2 1 a a (2) + 2 1 a a (3) + 1 a a (4) 1 a a
18. If A and B are acute angles satisfying 3sin2A + 2sin2B = 1 and 3 sin2A = 2sin2B, then cos( A + 2 B ) = (1) 0 (2) 1 (3) 1 2 (4) 1 4
19. q is in 3rd quadrant then ()4222 4sin sin cos 4 cos 42 πθ θ+θθ+−=
(1) 1 + 2sin q (2) 2 (3) 1 (4) 2 + 4sin q
20. If () 22 2sincos1cossin2, 2 xx π =−π
()π ≠+∈ 21,1 2 xnn then cos2x is equal to
(1) 1 5 (2) 3 5 (3) 4 5 (4) 1
21. The sum of the series, sin q sec3 q + sin3 q sec32 q + sin32 q sec32 q + .......... upto n terms, is
(1) θ−θ 1 1 tan3tan3 2 nn
(2) [tan3nq – tan q ] (3) θ−θ 1 tan3tan 2 n
(4) () θ− 1 tan31 2 n
22. The value of cot 24 π is:
(1) −−+ 2326
(2) 3236 (3) +++ 2326
(4) ++− 2326
23. = 0 1 tan7 2
(1) 2213 31 (2) + 13 13
(3) + 1 3 3 (4) + 21 2
24. If A + B + C = 0° then sinA + sinB + sinC =
(1) 4 sinA/2sinB/2sinC/2
(2) – 4 sinA/2sinB/2sinC/2
(3) 4 cosA/2cosB/2cosC/2
(4) – 4 cosA/2cosB/2cosC/2
25. A + B + C = 2S ⇒ sinS + sin(S – A) + sin(S – B) – sin(S – C) =
(1) 4coscoscos 222 ABC ⋅⋅
(2) 4coscossin 222 ABC ⋅⋅
(3) 4cossincos 222 ABC ⋅⋅
(4) 4sinsinsin 222 ABC ⋅⋅
26. If a , b are acute angles then 3cos21 cos2 3cos2 β− α= −β then
(1) tan a = 2tan b
(2) tan2tanα=β
(3) tan22tanβ=α
(4) tan b = 2tan a
27. If cos a + cos b = a, sin a + sin b = b and cos3 2,then cos θ α−β=θ= θ
(1) a2 + b2 – 3 (2) a2 + b2 – 2
(3) 3 – a2 + b2 (4) 22 4 ab +
28. cot16°cot44° + cot44°cot76° – cot76°cot16° is
(1) 0 (2) 1 (3) 3 (4) 4
29. If ()()()tantantan y xz == θ+αθ+βθ+γ then () sin2 xy xy + ∑α−β=
(1) 1 (2) –1
(3) 0 (4) none
30. If x = sin a + sin b , y = cos a + cos b then tan a + tan b =
(1) ()() ()222222 8 22 xy yxxyxy−+++−
(2) ()() ()222222 4 2 xy yxxyxy−+++−
(3) () ()2222 8 2 xy xyxy++−
(4) 4xy
31. If xy + yz + zx = 1, then 111222 y xz xyz ++= +++
(1) () () ()222 2 111xyz +−−
(2) () () ()222 2 111xyz −+−
(3) () () ()222 2 111xyz +++
(4) () () 222 2 11(1) xyz −++
32. Which of the following is not the value of sin27° – cos27° =
(1) 35 2 (2) 1sin54 −−°
(3) 51 22 (4) 1sin54−°
33. If a + b + γ = 2 q , then cos q + cos( q – a ) + cos( q – b ) + cos( q – γ ) =
(1) 4sincossin 222 αβγ ⋅⋅
(2) 4coscoscos 222 αβγ ⋅⋅
(3) 4sinsinsin 222 αβγ
(4) 3sin a sin b sin γ
Numerical Value Questions
34. If xsin3q + ycos3q = sin q cos q and xsin q = ycos q, then x2 + y2 is equal to _____
35. For 12,,.0, 2 n θθθπ …∈ , if ln(sec q 1 –tan q 1) + ln(sec q 2 – tan q 2) +....+ ln(sec q n – tan q n ) + ln π = 0 , then the value of |cos((secq1 + tanq1) (secq2 + tanq2).... (secq n – tan q n))| is equal to _____.
36. If 16 1 sin 5 a = and the value of 2248 1124 cos1sin1sin1sin aaaa +++ +++ is k, then 3 k =________, where [.] is greatest integer function.
37. a, b, γ∈ (0, π/4), such that −α−β−γ=−α+β+γ 1 (1tan)(1tan)(1tan)1(tantantan) 2 (1–tana)(1–tan b )(1–tan γ ) = 1–(tan a + tan b + tan γ ) then the value of tan( a + b + γ) is______.
38. If () ()() α+β−γγ=β≠γα−β+γβ tan tan , tan tan , then sin2 a + sin2 b + sin2 γ = ______.
39. Let A and B be non-zero real numbers such that 2(cosB – cosA) + cosAcosB = 1. Then −= 22 tan3tan 22 AB ______.
40. ππππ +++ 4444 235 cos cos cos cos 8888 πππ +++= 444 678 cos cos cos 888 ___.
41. If ()=+++ 2 23 sinsin3sin3 cos3 cos3cos3 n fxxxx xxx
()() 1 23 sin3,then/4/4 cos3 n n x ff x +π+π= ___.
42. ππ θ−θ− 3 16coscoscoscos 88 57 coscoscoscoscos4, 88 ππ
then the value of λ is
43. If πππ ++=− 2461 coscoscos 7772 and πππ =− 2461 coscoscos, 7778 then the numerical value of 22223 coseccoseccosec 777 πππ ++ must be
44. If 4sin27 =α+β , then sum of digits in ( a + b – ab + 2)4 =________.
45. 1 + cos2x + cos4x + cos6x – 4cosxcos2xcos3x = _________.
46. If the period of (() ) () cossin , tan/ nx nN xn ∈ is 6π, then n is_______.
47. If f(x) = 2(7cosx + 24sinx)(7sinx – 24cosx), for every x ∈ R , then maximum value of (f(x))1/4 is _______.
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. Wh at is the angle between the minutes hand and the hour hand of clock at 8:30?
2. If a circular wire of radius 7 cm is cut and bent again into an arc of a circle of radius 12 cm, then find the angle subtended by the arc at the centre of the circle.
3. Find the maximum value of cos(cos x).
4. If tan x + sec x = ex, then find the expression for cosx.
5. What is the range of f(x) = csc x?
6. What is the domain of f(x) = tan x?
48. 357 8sinsinsinsin _____. 14141414 ππππ =
49. The number of ordered pairs (x, y), when x, y ∈ [0, 10] satisfies
2 2 sec 1 sinsin21 2 xxy , is ____.
50. cos2q + cos2(60° + q ) + cos2(60° – q ) = k, then 4k = _____.
51. The value of cos61cos62 cos119 111 cos1cos2 cos59
52. Given that () 2sin2 cos2cos4 fn n θ=θ θ−θ ()()()() sin 23 , sinsin ffffn θ+θ+θ+…+θ=λθ
then the value of μ – λ, is ________.
53. In a triangle ABC, 2 sinsin1 , sin AAk A ++
then k = _____.
54. Minimum value of 9sec2θ + 4cosec2θ + 7 is = ____.
7. If 1 csccot 5 xx+= , then find the value of sin x.
8. What are the even trigonometric ratios?
9. If sincos 2 n xx π
and 0 < n < 7, then find the sum of the possible values of n.
10. If sin6 q + cos6 q = 1 – a sin2q + b sin4q + c sin6q, then find a + b + c .
11. What is the value of tan 15° + tan 75°?
12. If 2 1 22sin22tan 2 k π °= , then what is the value of 2 k ?
13. What is the value of tan 69° + tan 66° – tan 69° tan 66°?
14. If tanA + tanB + tanC = tanA tanB tanC, tan23,tan23 AC=−=+ , then find the measure of angle B.
15. What is the value of 11 tan22cot22 22 °+° ?
16. Find the value of () 22 4cos18cos54 °+° .
17. Find the simplest form of 4 sin 18° sin 54°.
18. What is the conjugate of 11 cot7tan7 22 °−° ?
19. What is the value of 22222 ++++ in terms of trigonometric ratios?
20. Find the sum form of –2 sin A sin B.
21. If cos x + cos y = a and sinx + siny = b, then find the expression for cos ( x + y).
22. Find the value of 4(cos 66° + sin 84°).
23. What is the period of sin ( ax + b)?
24. What is the period of tan ( ax + b)?
25. What is the period of constant function?
26. What is the maximum value of a cos x + b sin x + c?
Statement Type Questions
Each question has two statements: statement I (S-I) and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct,
(2) if both statement I and statement II are incorrect,
(3) if statement I is correct but statement II is incorrect,
(4) if statement I is incorrect but statement II is correct.
27. S-I : The measurement of angle 2 lies in the second quadrant.
S-II : The measure of 1 radian is approximately equal to 57 °.
28. S-1 : If tansec3 xx+= , then the value of x is 5 6 π
S-II : sec x + tan x , sec x – tan x are reciprocals to each other.
29. S-I : The domain of the function f(x) = tan x is the set of all real numbers other than odd multiples of right angles.
S-II : The graph of function f ( x ) = tan x intersect x-axis at all odd multiples of 2 π .
30. S-I : If q lies in the second quadrant, then 2 1sincosθθ−=−
S-II : If q lies in second quadrant, cos q is negative.
31. S-I : 1 + tan2 q= sec2q is valid for all real values of q.
S-II : An identity in terms of variable is an equation which is valid for all real values of x.
32. S-I : The domain of y = sin x and y = cos x are same.
S-II : The range of y = sin x and y = cos x are same.
33. S-I : The domain of y = sec x and y = tan x are same.
S-II : The range of y = sec x, y = tan x are same.
34. S-I : Let 0, 2 θπ ∈ and πθθ±=± sin sin, 2 n when n is even.
S-II : Let 0, 2 θπ ∈ and πθθ±=± sin cos, 2 n when n is odd.
35. S-I : If sin A + sin B + sin C = 3, then 2 ABC π ===
S-II : If cos A + cos B + cos C = –3, then A = B = C = π.
36. S-I : If A, B, C, D is a cyclic quadrilateral, then cosA + cosB + cosC + cosD =0.
S-II : If A, B, C, D is a cyclic quadrilateral, then sinA + sinB – sinC – sinD = 0.
37. S-I : If 2 sectan 3 xx+= , then measure of angle x lies in fourth quadrant.
S-II : If 1 csccot 5 θθ+= , then measure of angle q lies in second quadrant.
38. S-I : If a = sec x – tan x and b = csc x + cotx, then 1 1 b a b + = .
S-II : If a = sec x – tan x and b = csc x + cotx, then 1 1 ba a + =
39. S-I : If tan 15° + cot 195° = 2 a, then the value of 1 4 a a +=
S-II : tan195tan1523 °=°=+
40. S-I : If 1 tan 2 x = and 1 tan 3 y = , then 2(x + y) is right angle.
S-II : () tantan tan 1tantan xy xy xy + +=
41. S-I : tan 15° + cot 75° = 4
S-II : tan 15° + cot 15° = 4
42. S-I : cos 42° + cos 78° + cos 162° = 0
S-II : cos 162° = cos 18°
43. S-I : If cotcotcot3 ABC++= , then the triangle ABC is an equilateral triangle.
S-II : If 3, 1 xyzxyyzzx ++=++= , then x = y = z .
44. S-I : In a triangle ABC , if 2 cos 3 A =− , then cosB cosC are positive.
S-II : If A is acute angle, then all trigonometric ratios of angle A are positive.
45. S-I : In a triangle ABC, if C is acute angle, then tanA tanB > 1.
S-II : In a triangle ABC , if C is obtuse angle, then tanA tanB < 1.
46. S-I : tan50° – tan40° = 2 tan10°
S-II : tan70° – tan20° = 2 tan50°
47. S-I : 4 sin3 A + sin 3A = 3 sin A
S-II : 4 cos3 A – cos 3A = 3 cos A
48. S-I : 1 tan376432 2 °=−+−
S-II : 1 tan526432 2 °=+−−
49. S-I : The quadratic equation whose roots are sin2 18°, cos2 36° is 16x2 – 12x + 1 = 0.
S-II : The product of values of sin 36°, cos 36° is unity.
50. S-I : 2 sin 54° sin 66° 1 cos12 2 =°+
S-II : 2 sinA sinB = cos (A–B) – cos (A + B)
51. S-I : If f ( x ) is a periodic function with period p, then f ( x + p ) = f ( x ) for all x ∈ domain of f.
S-II : If f ( x ) is a periodic function with period p, then f(x – p) = f(x) for all x ∈ domain of f.
52. S-I : The period of sin4 x + cos4 x is right angle.
S-II : The fundamental period of of sin2 x + cos2 x is not defined.
53. S-I : The function cos x is not periodic.
S-II : The function cossin 2 x x π
is periodic.
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A),
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A),
(3) if (A) is true but (R) is false, (4) if both (A) and (R) are false.
54. (A) : The radius of the circle whose arc of length 15π makes an angle of 3 4 π radian at the centre is 20 cm.
(R) : The radian is measure of central angle subtended by an arc of length equal to the radius of the circle.
55. (A) : sin2 is positive and cos2 is negative.
(R) : The real number 2 is approximately equal to 114°, and it lies in the second quadrant.
56. (A) : The trigonometric ratios for an angle 2nπ+ q , n ∈ Z, are the same as those of q.
(R) : The measures of angles 2nπ+ q and q are coterminal angles.
57. (A) : cos 1 > cos 2
(R) : Cosine function is decreasing function in the first quadrant.
58. (A) : If sin x + csc x = 2, then the value of sinn x + cscn x is 2.
(R) : If 1 2 x x += , then the value of x is 1.
59. (A) : The function tan x is an odd function.
(R) : A function f ( x ) is said to be odd function if and only if f(–x) = –f(x).
60. (A): If the product of sin x and cos x is negative, then tan x is negative.
(R) : sin x and cos x are of different signs in second and fourth quandrants.
61. (A) : If A and B are complementary angles, then cos2 A + cos2 B = 1.
(R) : If A and B are complementary angles, then 2 AB π +=
62. (A) : The value of () sec1562 °=− .
(R) : cos(45° – 15°) = cos 45° sin15° + sin 45° cos15°.
63. (A) : 1 tan2221 2 °=−
(R) : If A + B = 45°, then (1+ tan A)(1 + tan B) = 2
64. (A) : cos11sin11 tan34 cos11sin11 °−° °= °+°
(R) : cossin tan 4cossin AA A AA π
−=
65. (A) : If 2 AB π += , then 2 tan (A – B) –tanA + tanB = 0.
(R) : If A and B are complementary angles, then tanA + tanB = 1
66. (A) : In a triangle ABC, if 3 cos 5 A =− and 7 sin 25 B = , then 3 tan 4 C =
(R) : In any triangle, there exists only one obtuse angle. Remaining two are acute angles.
67. (A) : tan70tan20 1 tan50 =
(R) : sin(A – B) = sinA cosB – cosA sinB
68. (A) : If A =35°, B = 15°, C = 40°, then cot A + cot B + cot C = cotA cotB cotC
(R) : If A + B + C is odd multiple of , 2 π then cotA + cotB + cotC = cotA cotB cotC.
69. (A) : cos(5 x ) = 16 cos 5 x – 20 cos 3 x + 5 cos x
(R) : cos(5x) = cos(3x + 2x) and cos 3x = 4 cos3 x – 3 cos x
70. (A) : 22 3 sinsin1 88 ππ +=
(R) : 22 3 cossin 88 ππ =
71. (A) : sin 3x = sin x(2 cos 2x + 1)
(R) : cos 2x = 2 sin2 x – 1
72. (A) : tan6° tan42° tan66° tan 78° = 1
(R) : tan q tan(60° + q )tan(60° – q ) = tan 3 q
73. (A) : The value of sin20° sin40° sin60° sin
80° is 3 16
JEE ADVANCE LEVEL
Multiple; Option Correct MCQs
1. The value of expression (α tan γ + b cot γ ) (αcot γ + b tan γ ) – 4α b cot22 γ depends on (1) α (2) b (3) γ (4) 0
2. If 2sec 2 A – sec 4 A – 2cosec 2 A + cosec 4 A =15/4, then tan A is equal to (1) 1 2 (2)
(R) : For any value of q, sinq + sin (60° + q)
() 1 sin60sin3 4 θθ°−=
74. (A) : 2 cosA cosB = sin (A + B) + sin(A – B)
(R) : sin(A ± B) = sinA cosB ± cosA cosB
75. (A) : –2sinA sinB = cos (A + B) – cos(A – B)
(R) : cos(A ± B) = cosA cosB ± sinA sinB
76. (A) : cosx + cos(240° + x) + cos(240° – x) = 0
(R) : cos ( A + B ) + cos( A – B ) = 2 cos A cosB
77. (A) : The period of cos (2x + 3) is π.
(R) : The period of cos x is 2π.
78. (A) : The period of sin x + cos x is 2π.
(R) : The period of af 1( x ) + bf 2( x ) is the least common multiple of periods of f1(x), f2(x).
79. (A) : The period of sin(π sinx) is twice the period of cos(π cosx).
(R) : The period of f(x) is f(x + p) = f(x) for all x ∈ domain of f(x)
80. (A) : The extremum value of a2 sin2x + b2 csc2x is 2ab.
(R) : The extremum value of a2 sec2x + b2 csc2x is (a + b)2.
3. If 3 tanA + 4 = 0, then the value of 2cot A – 5cosA + sinA is equal to
(1) π <<π 23 if 102 A
(2) π <<π 233 if 2 102 A
(3) π <<π 53 if 102 A
(4) π <<π 533 if 2 102 A
4. The value of ()ααα =−+ 2 cos2cotfec
αα + 2 cosec2cot can be (1) 2cotα
(2) – 2cotα
(3) 2
(4) –2
5. (m + 2)sin q + (2m – 1)cos q = 2m + 1, if
(1) 3 tan 4 θ = (2) 4 tan 3 θ =
(3) () 2 2 tan 1 m m θ = (4) () 2 2 tan 1 m m θ = +
6. If cotq + tanq = x and secq – cosq = y, then
(1) xsin q . cos q = 1
(2) sin2q= ycos q
(3) (x2y)1/3 + (xy2)1/3 = 1
(4) (x2y)2/3 – (xy2)2/3 = 1
7. If 2 3 sin2cos1, 25 yxx y + =++ + then the value of y lies in the interval
(1) 8 , 3 ∞
(3) 812 , 35
(2) 12 , 5 ∞
(4) 8 , 3 ∞
8. Four numbers n1, n2, n3 and n4 are given as n1 = sin15° – cos15°, n2 = cos93° + sin93°, n 3 = tan27° – cot27°, and n 4 = cot127° + tan127°. Then,
(1) n1 < 0 (2) n2 < 0
(3) n3 < 0 (4) n4 < 0
9. Let y = sin2x + cos4x. Then, for all real x, (1) the maximum value of y is 2
(2) the minimum value of y is 3 4 (3) y ≤ 1
(4) 1 4 y ≥
10. Suppose ABCD (in order) is a quadrilateral inscribed in a circle. Which of the following is/are always true?
(1) secB = secD
(2) cotA + cotC = 0
(3) cosecA = cosecC
(4) tanB + tanD = 0
11. If 11 cos 2 x x α =+ and 11 cos ,(0) 2 yxy y β =+> ,,,, xyR αβ∈ , then
(1) () sin sin R αβγγ∀γ ++=∈
(2) coscos1;, R αβ∀αβ=∈
(3) 2 (coscos)4;,R αβ∀αβ +=∈
(4) sin(α + b + γ ) = sinα + sin b + sin γ ; ,, R ∀αβγ ∈
12. If 44 sincos1 235 xx += , then
(1) 2 2 tan 3 x =
(2) 88 sincos1 827125 xx +=
(3) 2 1 tan 3 x =
(4) 88 sincos2 827125 xx +=
13. If 0 ≤ q ≤ π and 22 sincos 818130 θθ+= , then q is
(1) 30° (2) 60° (3) 120° (4) 150°
14. If A and B are acute angles such that sin A = sin2B, 2cos2A = 3cos2B, then
(1) A = π /6 (2) A = π /2
(3) B = π /4 (4) B = π /3
15. If 2cos223secθθ += , where q∈(0, 2π), then which of the following can be correct?
(1) 1 cos 2 θ = (2) tan q = 1
(3) 1 sin 2 θ =− (4) cot q = –1
16. Let f(x) = log(log1/3(log7(sinx + a))) define every real value of x . Then, the possible value of a is
(1) 3 (2) 4 (3) 5 (4) 6
17. If b > 1, sint > 0, cost > 0 and logb(sint) = x, then logb(cost) is equal to
(1) () 2 1 log1 2 x bb (2) 2log(1 – bx/2)
(3) 2 log1 x bb (4) 2 1 x
18. If x = sec φ – tan φ and y = cosec φ + cot φ , then
(1) 1 1 xy y + = (2) 1 1 xy y = +
(3) 1 1 x y x + = (4) xy + x – y + 1 = 0
19. If x = acos3q sin2q , y = asin3q cos2q, and ()() 22 , () p q xy pqN xy + ∈ is independent of q, then
(1) p = 4 (2) p = 5 (3) q = 4 (4) q = 5
20. For 0 < φ < π/2, if 22 00 cos,sinnn nn xy ∞∞
and 22 0 cossinnn n z ∞ φφ = =∑ , then xyz =
(1) xy + z (2) xz + y (3) x + y + z (4) yz + x
21. A circle centred at O has radius l and contains point A. Segment AB is tangent to the circle at A and AOB ∠θ = . If point C lies on OA, and BC bisects the angle ABO, then OC equals
(1) sec q (sec q – tan q )
(2) 2 cos 1sin θ +θ
(3) 1 1sin+θ
(4) 2 1sin cos θ θ
22. Which of the following is/are correct?
(1) ()()()lnsin lnsin (tan)(cot),0,/4 xx xxx∀π>∈
(2) () ln ln 45,0,/2 cosecxcosecx x ∀π<∈
(3) ()()()lncos lncos (1/2)(1/3),0,/2 xx x ∀π<∈
(4) ()()() lntanlnsin, 220,/2 xx x ∀π>∈
23. If tanq + tanφ = a and cotq + cotφ = b, and q – φ = a≠0, then
(1) ab > 4 (2) ab = 4
(3) α=() + 2 2 4 tan () abab ab
(4) () + α= 2 2 4 cot () abab ab
24. If 3 sin b = sin(2 a + b ), then tan( a + b ) –2tan a is (1) independent of a (2) independent of b (3) dependent on both a and b (4) independent of both a and b
25. If 0 ≤ x, y ≤ 180° and ()()−=+= 1 sincos, 2 xyxy then the values of x and y are given by
(1) x = 45°, y = 15° (2) x = 45°, y = 135° (3) x = 165°, y = 15° (4) x = 165°, y = 135°
26. Suppose cosx = 0 and ()+= 1 cos 2 xz . Then the possible value(s) of z is (are).
(1) π 6 (2) π 5 6 (3) π 7 6 (4) π11 6
27. If sinx + 7 cosx = 5, then () 1 cos , 2 x −φ= where
(1) φ= 7 cos 50 (2) φ= 7 cos 75
(3) φ= 1 sin 50 (4) φ= 26 sin 75
28. In cyclic quadrilateral ABCD, if == 3 12 cotandtan 45 AB , then which of the following is (are) correct?
(1) = 12 sin 13 D
(2) () 16 sin 65 AB+=
(3) = 15 cos 13 D
(4) ()+= 16 sin 65 CD
29. The value of x in (0, π /2) satisfying the equation −+ += 3131 42 sincosxx is
(1) π 12 (2) π 5 12
(3) π 7 24 (4) π11 36
30. If π α=<α<π 3 sin, 52 and −πβ=π<β<53 cos, 132 then the correct statements is/are
(1) ()α−β= 63 tan 16
(2) ()α+β= 33 tan 56
(3) α= 24 sin2 25
(4) β= 119 cos2 169
31. In ∆ABC, if tanB + tanC = 5 and tanAtanC = 3, then
(1) ∆ ABC is an acute angled triangle (2) ∆ ABC is an obtuse angled triangle
(3) sum of all possible values of tan A is 10 (4) sum of all possible values of tan A is 9
32. If ()−= 3 cos 5 AB and tanAtanB = 2, then
(1) = 1 coscos 5 AB
(2) = 1 coscos 5 AB
(3) = 2 sinsin 5 AB
(4) = 2 sinsin 5 AB
33. If (1 + tan a )(1 + tan4 a ) = 2, then a = (1) π
34. If tan(2 a + b ) = x and tan( a + 2 b ) = y , then [tan3( a + b )][tan( a – b )] is equal to (wherever defined)
(1) + 22 22 1 xy xy (2) + 22 22 1 xy xy
(3) + + 22 22 1 xy xy (4) 22 22 1 xy
35. For <θ<π 0 2 , the solution (s) of
36. Assume that a , b , γ satisfy 0 < a < b < γ < 2π. If cos(x + a)+ cos(x + b) + cos(x + γ) = 0 for all x ∈ R, then which of the following is/are correct? (1) α−β=−π 2 3 (2) γ−β=π 2 3 (3) α−β=−π 3 (4) γ−β=π 3
37. If ()()() β−γ+γ−α+α−β=− 3 coscoscos 2
)() β−γ+γ−α+α−β=− 3 coscoscos 2 , then
(1) ∑ cos a = 0
(2) ∑ sin a = 0
(3) ∑ cos a sin a = 0
(4) ∑ (cos a+ sin a ) = 0
38. If ()=≠ tan3 ;1 tan Akk A , then
(1) = 2 cos1 cos32 Ak Ak (2) 2 cos31 cos2 Ak Ak =
(3) < 1 3 k (4) k > 3
39. Let f : (–1, 1) → R be such that ()θ= −θ 2 2 cos4 2sec f for 0,,. 442
Then, the value(s) of
1 3 f is/are (1) 3 1 2 (2) + 3 1 2 (3) 2 1 3 (4) +
40. If cosb is the geometric mean between sina and cos a , where 0 < a , b < π /2, then cos2 b is equal to (1)
2sin2 4 (2)
41. If cosx = sin a cot b , sinx = cos a, then the value of tan(x/2) is
(1) –tan( a /2)cot( b /2)
(2) tan( a /2)tan( b /2)
(3) –cot( a /2)tan( b /2)
(4) cot( a /2)cot( b /2)
42. If (x – a)cosq + ysinq = (x – a)cosφ + ysinφ = a and tan( q /2) – tan( φ /2) = 2b, then
(1) y2 = 2ax – (1 – b2)x2
(2) θ() =+ 1 tan 2 ybx x
(3) y2 = 2bx – (1 – a2)x2
(4) φ() =− 1 tan 2 ybx x
43. If () 35 2sinsin2sinsin 2222 n f θθθθθ=+ 7 2sinsin 22 θθ ++ ..... + () θθ +∈2sinsin21, 22 nnN then which of the following is/are correct?
(1) π = 9 1 4 2 f
(2) π =∈ 2 0, n fnN n
(3) 5 1 2 f π =
(4) 9 51 4 2 f π =−
44. Let P = sin25° sin35° sin60° sin85° and Q = sin20° sin40° sin75° sin80°. Which of the following relation(s) is (are) correct?
(1) P + Q = 0 (2) P – Q = 0
(3) P2 + Q2 = 0 (4) P2 – Q2 = 0
45. Let f(x) = a1cos( a 1 + x) + a2cos( a 2 + x) + ..... + a ncos( a n + x). If f(x) vanishes for x = 0 and x = x1 (where x1≠ kπ, k ∈ Z), then
(1) a1cos a 1 + a2cos a 2 + ..... + a n cos a n = 0.
(2) a1sin a 1 + a2sin a 2 + ..... + a n sin a n = 0.
(3) f(x) = 0 has only two solutions 0, x1
(4) f(x) is identically zero ∀ x.
46. Which of the following quantities are rational?
(1)
115 sinsin 1212
(2)
94 sec 105 cosec
(3)
(4)
+
44 sincos 88
248 1cos1cos1cos 999
47. The expressions (tan4x + 2tan2x + 1)cos2x, when x = π /12, can be equal to (1) ()423 (2) () + 421 (3) 16cos2π /12 (4) 16sin2π /12
48. If cot3a + cot2a + cot a = 1, then
(1) cos2 a .tan a = –1
(2) cos2 a .tan a = 1
(3) cos2 a – tan2 a = 1
(4) cos2 a – tan2 a = –1
49. Let ()()
2141 1cos 1cos 44 kk kk
then,
(1) ()= 1 3 16 P (2) ()= 22 4 16 P
(3) ()= 35 5 32 P (4) ()= 23 6 16 P
50. Which of the following identities, wherever defined, hold(s) good?
(1) cot a – tan a = 2cot2 a
(2) tan(45° + a) – tan(45° – a) = 2cosec2a
(3) tan(45° + a ) + tan(45° – a ) = 2sec2 a
(4) tan a + cot a = 2tan2 a
51. The equation 3 33 48 xx−=− is satisfied by (1) π
=
(3) π = 23 cos 18 x (4) π =−
7 sin 18 x
52. If =− tan sin 2 x cosecxx , then
tan2 2 x is equal to
(1) 25 (2) 52
(3) () ()−+ 94525
(4) () ()+− 94525
53. If 1sin41 1sin41 A y A −+ = +− , then one of the value of y is
(1) –tanA (2) cotA
(3) tan 4 A π +
−+
(4) cot 4 A π
54. If sin(x + 20°) = 2sinx cos40°, where x∈(0, π /2), then which of the following hold(s) good?
(1) cosx = 1/2
(2) cosec4x = 2
(3) sec62 2 x =−
(4) ()tan23 2 x =−
55. If p = sin(A – B)sin(C – D), q = sin(B – C) sin(A – D), r = sin(C – A)sin(B – D), then
(1) p + q – r = 0 (2) p + q + r = 0
(3) p – q – r = 0 (4) p3 + q3 + r3 = 3pqr
56. If 3sin b = sin(2 a + b ), then tan( a + b ) –2tan a is
(1) independent of α (2) independent of β (3) dependent of both α and β
(4) independent of both α and β
57. If x = sin( a – b )sin( γ – δ ), y = sin( b – γ ) sin(a – δ) and z = sin(γ – a)sin(b – δ) then
(1) x + y + z = 0 (2) x + y – z = 0
(3) y + z – x = 0 (4) x3 + y3 + z3 = 3xyz
58. For α = π/7, which of the following hold(s) good?
(1) tan a tan2 a tan3 a = tan3 a – tan2 a –tan a
(2) cosec a = cosec2 a + cosec4 a
(3) cos a – cos2 a + cos3 a = 1/2
(4) 8cos a cos2 a cos4 a = 1
59. If cosx + cosy = a, cos2x + cos2y = b, cos3x + cos3y = c, then
(1) 22 coscos1 2 b xy+=+
(2) () 2 2 coscos 24 ab xy + ⋅=−
(3) 2a3 + c = 3a(1 + b)
(4) a + b + c = 3abc
60. If () () () 7 coscos2cos.cos32 222 7,sinsin2sin.sin32 222 n n f n θθθ +θ++…+− θ= θθθ +θ++…+− then
(1) 3 3 21 16 f π =−
(2) 5 21 28 f π =+
(3) 7 23 60 f π =+
(4) none of these
61. The value of the expression 248 tan2tan4tan8cot 7777 ππππ +++ is equal to (1) ππ + 22 coseccot 77
(2) tancot 1414 ππ
(3) 2 sin 7 2 1cos 7 π π (4) 2 1cotcos 77 2 sinsin 77 ππ ++ ππ +
62. If xcos a + ysin a = xcos b + ysin b 20, 2 a π =<αβ< , then
(1) 22 4 coscos ax xy α+β= +
(2) 22 22 4 coscos ay xy αβ= +
(3) 22 4 sinsin ay xy α+β= +
(4) 22 22 4 sinsin ax xy αβ= +
63. If sin a + sin b = l, cos a cos b = m, and ()tantan1 22 n αβ =≠ , then
(1) () 22 2 cos 2 lm α−β=+−
(2) () 22 22 cos ml ml α+β= +
(3) 22 1 12 nlm nn ++ =
(4) if 2 lm π α+β==
64. sincos Iftan ,then sincos θ=α−α α+α
(1) sincos2sin α−α=±θ
(2) sincos2cos α+α=±θ
(3) cos2 q = sin2 a
(4) sin2 q + cos2 a = 0
65. coscossinsin sinsincoscos nn ABAB ABAB ++ + (n:even or odd) is equal to
(1) 2tan 2 nAB
(2) 2cot 2 nAB
(3) 0 (4) None of these
66. If a function f(n) is defined from N → R (i.e., set of natural numbers to real numbers) such that 1 ()sincos n r fnrr nn ππ =
r ∈ N, and n ≥ 1, then which of the following option(s) is/are correct?
(1) f(n) contains only positive integers in its range.
(2) Number of real values of x, for which 2f(x) + 1 = 0, is 0.
(3) f(n) contains exactly one non-positive integer in its range.
(4) ()42 f =
67. Which of the following function(s) is/are periodic?
(1) ()[] 2 2 x fxx = where [ ] denotes greatest integer function
(2) g ( x ) = sgn { x } where { x } denotes the fractional part function
(3) h(x) = sin–1(cos(x2))
(4) ()() = 1 cossin kxx
68 Let f(x) = absinx + 2 1cos,1baxca−+< , b > 0
(1) maximum value of f(x) is b if c = 0
(2) difference of maximum and minimum values of f(x) is 2b
(3) f(x) = c, if x = –cos–1a
(4) f(x) = c, if x = cos–1a
69. In ∆ ABC, which of the following is true?
(1) 33 sin.sin.sin 8 ABC ≤
(2) 222 9 sinsinsin 4 ABC++≤
(3) sinA.sinB.sinC is always positive.
(4) sin2 A + sin2 B ≤ 1 + cos C
70. If 1 5cos12sin a xx = + , then for all real x
(1) 1 13 a ≤
(2) the least positive value of a is 1 13
(3) 11 1313 a ≤≤
(4) the greatest negative value of a is –1 13
=+∀∈
71. Given () 4 4 1 tan2tan 0, 4 tan fxxx x π
and. If the value of f(sinx) + f(cosx), when 4 x π = , is k, then which is/are correct?
(1) k is divisible by 7.
(2) k is divisible by 4.
(3) Tens place of k is 9.
(4) k is even.
72. If 44 5 1 sincossin21 2 a xxx = +−+ . then a can be
(1) 2 (2) 3 (3) 4 (4) 5
Numerical/Integer Type Questions
73. If tan3q + cot3q = 52, then tan2q + cot2q = λ , where 2 λ equals to ____.
74. If S = () 89 2 1 1 1tan n n =° + ∑ , then 11 s is____.
(where [.] denotes GIF)
________.
75. If x = a sec n q and y = b tan n q , then 2 2 nnxy ab
76. If 2tanq = secq, then 4cot2q – 3tan2q =_____.
77. If p cosec q + q cot q = 2 and p 2 cosec 2 q –q2cot2q = 5, then the value of 22 81 pq is ______.
78. If 0 4 x π << and 5 cossin 4 xx+= , then the value of 16(cosx – sinx)2 is____.
79. Suppose, for some angles x and y , the e qu ations 22 3 sincos 2 a xy+= and 2 22 cossin 2 a xy+= hold simultaneously. The possible value of a is ___.
80. If secAtanB + tanAsecB = 91, then the value of (secAsecB + tanAtanB)2 – 912 is equal to ____.
81. The value of the expression (2sin 2 91° –1)(2sin 292° – 1)...(2sin 2180° – 1) is equal to____.
82. 2sin1cossin 1cossin1sin θθθ θθθ −+ +++ is equal to____.
83. Suppose A and B are two angles such that (),0,AB ∈π , and they satisfy sinA + sinB = 1 and cosA +cosB = 0. Then, the value of 12 cos 2A + 4 cos 2B is ____.
84. If () = ∑+=− 88 o2 1 tanran1cot1, t oo r rK then K = ____.
85. let 0 ≤ a, b, c, d ≤π where b and c are not complementary such that 2 cos a + 6 cos b + 7 cos c + 9 cos d = 0
2 sin a – 6 sin b + 7 sin c – 9 sin d = 0 If () () + = + cos cos adm bcn , where m and n are relatively prime positive integers then the value of m – n = _____.
86. tan15° + cot15° = ____
87. Let a and b be real numbers such that −<β<ππ <α< 0 44 If ()α+β= 1 sin 3 and ()α−β= 2 cos 3 then the greatest integer less than or equal to 2 9sincoscossin 4cossinsincos
αβαβ +++ βαβα is ___.
88. If A + B + C = π , then the value of ++ coscoscos sinsinsinsinsinsin ABC BCACAB is ___.
89. If α=β= ++ 1 tan,tan 121 m mm , then the value of a + b is π π+∈ , nnZ k . Then, the value of k is____.
90. πππ ++ cotcotcot22223 777 is equal to ____.
91. If cos5q = acosq + bcos3q + ccos5q + d, then ++ = 3 acd _____.
92. If a = 2π/13 then, 4(cos2a + cos22a + cos23a + cos24a+ cos25a+ cos26a) – 7=_______.
93. If °−°=cosec103sec10 k , then k equal to ____.
94. The exact value of the expression +−
sin40sin80sin20 sin80sin20sin40 ______.
95. For 2 13 π α= , 4|cos a cos5 a + cos2 a cos3 a + cos4 a cos6 a | = ____.
96. If k1 = tan 27 q – tan q and 2 sinsin3sin9 cos3cos9cos27 k θθθ =++ θθθ then 1 2 k k = ___.
97. ()()sin22tan Ifsin then 5 tan ABAB B A ++ == ____.
98. If A + B + C = π and sin2sin2sin2 sinsinsin sinsinsin222 ABCABC ABC ++=λ ++ , then the value of λ must be ___.
99. () () () () 6coscos3sin8sin2 ___. sin5sincos4cos6 α−αα+α = α−αα−α
100. sin212° + sin221° + sin239° + sin248° – sin29° – sin218° = ____.
101. The number of negative integers in the range of the function ()() 2 cossinsin3fxxxx =++ is ____.
102. sin2x + 4sinx + 5 ∈ [K, 5K] ⇒ K =____
103. The sum of the maximum and minimum values o f sin2222 sin 33 ππθθ ++−
is______.
104. If the period of (() ) cossin , tan nx nN x n ∈ is 6π, then n = ____.
105. If 3cos5cos3 3 ab π θθ ≤+++≤ , then the value of () 2 ba is
106. If 11 cos3 248yx π =−− , then the period of 'y' is π.
107. The minimum value of 2cosx – 3cos2x + 5 is ____.
108. The maximum value of sin2 x + 2sin x + 3 is _____.
109. The minimum value of (sin q + cosec q )2 + (cos q + sec q )2 is ___.
Passage-based Questions
(Q: 110 – 112)
Let f( q ) = sin q – cos2 q – 1, where R θ∈ and m ≤ f( q ) ≤ M.
110. Let N denote the number of solution of the equation f(q) = 0 in [0, 4π]. Then the value of () 22 1 loglog 1 mm N N + + is equal to (1) 1 2 (2) 1 (3) 1 2 (4) –1
111. The value of (4m + 13) is equal to (1) 0 (2) 4 (3) 5 (4) 6
112. Sum of the all values of x satisfying the equation 111 , x mmm ∞=+++……… is (1) 1 3
(Q: 113 – 114)
The method of eliminating ‘q’ from two given equations involving trigonometrical function of ‘ q ’. By using given equations involving ‘ q ’ and trigonometrical identities, we shall obtain an equation not involving ‘ q ‘. On the basis of above information answer the following questions.
113. If xsin3 q + ycos3 q = sin q cos q and xsin q –ycos q = 0, then (x, y) lies on (1) a circle (2) a parabola (3) an ellipse (4) a hyperbola
114. If cossin xy abθθ = and 22 , cossin axbyab θθ ==− then (x, y) lies on
(1) a circle (2) a parabola (3) an ellipse (4) a hyperbola
(Q: 115 – 116)
In ∆ ABC, BC = 1, sin,sin,cos123 222 ABA xxx === , and
115. Length of side AC is equal to (1) 1/2 (2) 1 (3) 2 (4) can’t be determined
116. If 90 A ∠= , then area of ∆ ABC is (1) 1/2 sq. units (2) 1/3 sq. units (3) 1 sq. units (4) 2 sp. units
(Q: 117 – 118)
If a, b, and c are the sides of ∆ ABC such that 222222 2 22 32.330 aabcbc +++ −+= then
117. Triangle ABC is (1) equilateral (2) right angled (3) isosceles right angled (4) obtuse angled
118. If sides of ∆ PQR are a , b sec C , cosec C . Then, area of triangle is ____ units.
(1) 2 3 4 a (2)
(Q: 119 – 120)
The method of eliminating q from two given equations involving trigonometrical function of q. By using given equations involving q and trigonometrical identities, we shall obtaining an equation not involving q. On the basis of the the above information, answer the following questions.
119. If tanq + sinq = m and tanq – sinq = n then (m2 – n2)2 is
(1) 4 nm (2) 4mn
(3) 16 mn (4) 16mn
120. If s in q + cos q = a and sin 3 q + cos 2 q = b , then we get λ a 3 + m b + n a = 0 when are independent of q , then the value λ 3+m 3+ n 3 is
(1) –6 (2) –18 (3) –36 (4) –98
(Q: 121 – 123)
Let f(x) = sin6x + cos6x + k(sin4x + cos4x) for some real number k
121. Value of k, for which f(x) is constant for all values of x, is (1) –1/2 (2) 1/2 (3) 1/4 (4) –3/2
122. All real numbers k for which the equation f(x) = 0 has solution lie in
(1) [–1, 0] (2) 1 0, 2
(3) 1 1, 2 (4) none of these
123. Number of values of k, for which f(x) = 0 is an identity, is
(1) 0 (2) 1
(3) infinite (4) None of these
(Q: 124 – 126)
a , b , γ , and δ are angles in I, II, III, and IV quadrant respectively, and not one of them is an integral multiple of π /2. They form an increasing arithmetic progr ession.
124. Which of the following holds?
(1) cos( a + δ ) > 0
(2) cos( a + δ ) = 0
(3) cos( a + δ ) < 0
(4) Data is insufficient
125. If a + b + γ + δ = q and a = 70°,
(1) 400° < q < 580° (2) 470° < q < 650°
(3) 680° < q < 860° (4) 540° < q < 900°
126. Which of the following does not hold?
(1) sin( b + γ ) = sin( a + δ )
(2) sin( b – γ ) = sin( a – δ )
(3) tan2( a – b ) = tan( b – δ )
(4) cos( a + γ ) = cos2 b
(Q: 127 – 130)
If sin a = A sin( a + b ), A ≠ 0, then answer the following question.
127. The value of tan a is (1) β −β sin 1cos A A (2) β +β sin 1cos A A
(3) β −β cos 1sin A A (4) β +β cos 1cos A A
128. The value of tan b is
(1)
() α+β αβ sin1cos coscos A A
(2) () α−β αβ sin1cos coscos A A
(3)
() α−β αβ cos1sin coscos A A
(4) () α+β αβ cos1sin coscos A A
129. Which of the following is the value of tan( a + b )?
(1) β β− sin sin A (2) αα β−α 2 sincos cossin A
(3) αα β+α 2 sincos cossin A (4) α α− sin cos A
(Q: 130 – 131)
In a ∆ ABC, if = 31 coscoscos 8 ABC and + = 33 sinsinsin 8 ABC , then
130. −= tantantan23___ ABC
131. ++−= tanantantantantan43___ AtBBCCA
(Q: 132 – 134)
Let a , b , γ , δ and be the solutions of the equation π θ+=θ
tan3tan3 4 , no two of which have equal tangents.
132. The value of tan a + tan b + tan γ + tan δ is (1) 1/3 (2) 8/3 (3) –8/3 (4) 0
133. The value of tan a tan b tan γ tan δ is (1) –1/3 (2) –2 (3) 0 (4) none of these
134. The value of +++ αβγδ 1111 tantantantan is (1) –8 (2) 8 (3) 2/3 (4) 1/3
(Q: 135 – 136)
If 7 q = (2n + 1) π , when n = 0, 1, 2, 3, 4, 5, 6, then, on the basis of the above information, answer the following q uestions.
135. The equation whose roots are cos π /7, cos3 π /7, and cos5 π /7, is
(1) 8x3 + 4x2 + 4x + 1 = 0
(2) 8x3 – 4x2 – 4x + 1 = 0
(3) 8x3 – 4x2 – 4x – 1 = 0
(4) 8x3 + 4x2 + 4x – 1 = 0
136. The value of sec π /7 + sec3 π /7 + sec5 π /7 is (1) 4 (2) –4 (3) 3 (4) –3
3: Trigonometric Functions
(Q: 137 – 138)
137. πππ = 91113 8sinsinsin_______ 141414 138. ()πππ
(Q: 139 – 140)
If a = sin10°, b = sin50°, c = sin70° then answer the following
139. The value of +
8
c is equal to_____.
140. The value of +− 111 abc is equal to ______.
(Q: 141 – 142)
Let a and b are positive integers such that π
832768cos a b then answer the following.
141. The value of a is ______.
142. The value of a + b is______.
(Q: 143 – 144)
The roots of unity can be taken as vertices of a regular polygon. One can make use of this interpretation to derive interesting identities. Use this idea to answer the two questions given below.
143. The value of sin2°sin4° .... sin(2k)° ... sin90° is λ 44 2 , where λ∈ N. Then, λ =_____.
144. The value of sin1°sin3° ... sin(2 k – 1)° ... sin179° is 1 2 k , where k ∈ N. Then k =___.
(Q: 145 – 147)
If 11 sinsin and coscos 43 α+β=α+β= , then answer the questions given below.
145. The value of sin( a + b ) is (1) 24 25 (2) 13 25 (3) 12 13 (4) none of these
146. The value of cos( a + b ) is (1) 12 25 (2) 7 25 (3) 12 13 (4) none of these
147. The value of tan( a + b ) is (1) 25 7 (2) 25 12 (3) 25 13 (4) 24 7
(Q: 148 – 149)
If the angles α, β, and γ of a triangle satisfy the relation 33 sinsinsin 2222 α−βα−γα
. Then answer the following questions.
148. The measure of the smallest angle of the triangle is (1) 30° (2) 40° (3) 45° (4) 50°
149. Triangle is
(1) an acute angled
(2) right angled but not isosceles
(3) isosceles
(4) isosceles right angled
(Q: 150 – 151)
If 3 coscos 2 α+β= and 1 sinsin 2 α+β= and q is arithmetic mean of α and β.
150. sin2 q + cos2 q = _______.
151. cos( a – b ) = ____________.
(Q: 152 – 153)
Let f(x) = 3cosx + 4sinx + 15
152. The maximum value of f(x) is _____.
153. The minimum value of f(x) is ___.
(Q: 154 – 156)
The maximum and minimum values of a cos q± b sin q + c are 22cab ++ and 22cab −+ respectively i.e.,
cossin cababccab θθ −+≤±+≤++
154. The maximum and minimum values 7cos q + 24sin q = (1) 25 and –25 (2) 24 and –24 (3) 5 and –5 (4) 10 and –10
155. The value of 5cos3cos3 3 θθπ
+++
lies between (1) –4 and 10 (2) 4 and 10 (3) –4 and –1 (4) 4 and –10
156. If 3cos5sin 6 b αθθπ ≤+−≤ then ' a ' and 'b' are
(1) 19,19
(2) 19,19
(3) 19,19 (4) 19,19
(Q: 157 – 159)
If f1(x) and f2(x) are periodic functions with periods T1 and T2 respectively, then we have the period of h ( x ) = f 1 ( x ) + f 2 ( x ) = LCM
of { T 1, T 2} : If h ( x ) is not an even function (or) LCM of {T1, T2 } : If f1(x) and f2(x) are complementary function and even.
157. Let f(x) = cot2x, g(x) = cotx, then the period of f(x) . g(x) is
(1) π (2) 2 π (3) 2 π (4) 2
158. If ()()cot,sin2 2 x fxgxx π =π = , then the period of f(x). g(x) is (1) π (2) 2 π (3) 2 π (4) 2
159. If f(x) = cosec2x, g(x) = cosec 2x, then the period of f(x).g(x) is (1) π (2) 2 π (3) 2 π (4) 2
Matrix Matching Questions
160. If sincostanxxxk abc === , then match the items of List I with the items of List-II and choose the correct match.
List I List II
(A) bc (p) 22 1 bk
(B) a2 + b2 (q) 1 ak
(C) 1 1 ak ckbk + + (r) a k
(D) a2 + b2+c2 (s) 2 1 k
(A) (B) (C) (D)
(1) r s q p
(2) s p r q
(3) p q s r
(4) q r p s
161. Match the items of List-I with items of List-II and choose the correct option
List I
List II
(A) If a = xcos2a + ysin2a , then (x – a)(y – a) + (x + y)2sin2α cos2α = (p) 2
(B) If x = cot q + tan q ; y = sec q – cos q, then (x2y)2/3 – (xy2)2/3 = (q) 2a2/3
(C) If n = asin3q + 3a cos2q sin q and m = acos3q + 3a cos q sin2q, then (m+n)2/3 + (m–n)2/3 = (r) 1
(D) cossin1xy ab θθ+= and sincos1,xy ab −=θθ then 2 2 22 xy ab += (s) 0
(A) (B) (C) (D)
(1) s r q p
(2) p r q s
(3) s q r p
(4) r s q p
162. Match the items of List-I with items of List-II and choose the correct option.
List I
List II
(A) 3 tanx + 27 cotx ≥ (x ∈ Q1) (p) 24
(B) 5sec2x + 125 cos2x ≥ (q) 18
(C) 16 cosec2x + 9sin2x ≥ (r) 50
(A) (B) (C)
(1) p q r
(2) r p q
(3) q r p
(4) r q p
163. Match the items of List-I with the items of List-II and choose the correct option.
List I
List II
(A) Least value of 3cos2q + 4sin2q (p) 1
(B) If A > 0, B > 0, and 3 AB π += , then maximum value of tanA tanB (q) 0
(C) The value of 6(sin6q + cos6q ) – 9(sin4q + cos4q ) + 4 (r) 2
(D) If A+B+C = π and cosA = cosB cosC, then value of tanB tanC is (s) 1/3 (t) 3
(A) (B) (C) (D)
(1) r t p s
(2) p s r q
(3) t s p r
(4) r q p s
164. If 4 1 cos – sin , 0 5 θθ=<θ<π , then match List-I with items of List-II and choose the correct option.
List I
List II
(A) cos q + sin q (p) 4 5
(B) sin2 q (q) 7 5
(C) cos2 q (r) 24 25
(D) cos q (s) 7/25
(A) (B) (C) (D)
(1) p q r s
(2) q r s p
(3) r s p q
(4) s p q r
165. Match the items of List-I with the items of List-II and choose the correct option.
List I
(A)
List II
If 2 AB π += , then tanB + 2tan(A–B) (p) 2tanA tanB
(B) If 4 AB π += , then (1+tanA)(1+tanB)= (q) tanA
(C) If 5 4 AB π += , then cotcot (1cot)(1cot) AB AB = ++ (r) 2
(D) If 3 4 AB π += , then (1+tanA)(1+tanB)= (s) 1 2
(A) (B) (C) (D)
(1) p q r s
(2) q r s p
(3) r s p q
(4) s p q r
166. Match the items of List-I with the items of List-II and choose the correct option.
List I
(A) cot.cot 44
+−
List II
(p) () 1 1sin2 2 +θ
(B) sin(45°+ q ) cos(45°– q ) (q) tan56°
(C) cos11sin11 cos11sin11 °+° °−° (r) 3 2
(D) sin2 75° – sin2 15° (s) 1
(A) (B) (C) (D)
(1) p q s r
(2) p s q r
(3) s p q r
(4) p q r s
167. Match the items of List I with the items of List II and choose the correct option.
List I
List II
(A) sin(410°–A)cos (400° + A) + cos (410° – A) sin (400° + A) = (p) –1
(B) 22 cos1cos2 2sin3sin1 °−° °° (q) 0
(C) sin(–870°)+cosec (–660°)+ tan(–855°) + 2 cot(840°) + cos (480°) + sec (900°) (r) 1 2
(D) If 4 cos 5 θ = where 3 ,2 2 π θπ ∈ and 3 cos 5 φ = where 0, 2 φπ ∈ then cos( q – φ )= (s) 1 (t) 2
(A) (B) (C) (D)
(1) s r p q
(2) s r t p
(3) s p r t
(4) r s p q
168. Match the items of List I with the items of List II and choose the correct option.
List I List II
(A)
ππ +θ−−θ= tantan 44 (p) 1
(B)
(C)
(D)
ππ +θ++θ= tant 44 co (q) 4sec22q
ππ +θ−θ= tan.tan 44 (r) 2sec2q
ππ +θ−θ= 22 coscos 44 ecec (s) 2tan2q
(A) (B) (C) (D)
(1) s r p q
(2) p s q r
(3) q p r s
(4) r q s p
169. Match the items of List I with the items of List II and choose the correct option.
List I List II
(A) +− 000 cos20cos803cos50 (p) –1
(B) πππ +++ 0 23 cos0coscoscos 777 πππ +++ 456 coscoscos 777 (q) 3 4
(C) cos20° + cos40° + cos60°–4cos10°cos20°cos30° (r) 1
(D) cos20°cos100° + cos100° cos140° –cos140°cos200° (s) 0
(A) (B) (C) (D)
(1) s r p q
(2) r s p r
(3) r s q p
(4) p r q s
170. Match the items of List I with items of List II and choose the correct option.
List I List II
(A) In ∆ ABC, if cos2A + cos2B+ cos2C = –1 then we can conclude that triangle is
(B) In ∆ ABC, if tanA > 0, tanB > 0 and tanAtanB < 1, then triangle is
(C) In ∆ ABC, if cos3 A + cos3 B + cos C = 3cosAcosBcosC then triangle is
(D) In ∆ ABC cot A > 0, cot B > 0 and cot Acot B < 1, then triangle is
(p) Equilateral triangle
(q) Right angle triangle
(r) Acute angle triangle
(s) Obtuse angle triangle
(A) (B) (C) (D)
(1) q s p r
(2) q p r s
(3) r s q p
(4) r s p q
171. Match the items of List I with items of List II and choose the correct option.
List I List II
(A) In triangle ABC, if 3sin A + 4cosB = 6 and 3cos A+4sinB = 1, then C∠ can be (p) 60°
(B) In any triangle, if (sinA+ sinB + sinC) (sinA + sinB – sinC) = 3sinAsinB, then the angle C∠ is (q) 30°
(C) If 8 sin x cos5 x –8sin5 x cos x = 1, then x is (r) 165°
(D) ‘O’ is the center of the inscribed circle in a 30,60,90 ∠°∠°∠° triangle ABC with right angled at C.If the circle is tangent to AB at D, then the angle COD∠ is (s) 7.5°
(A) (B) (C) (D)
(1) p q r s
(2) r s p q
(3) q p s r
(4) r q p s
172. Match the items of List I with the items of List II and choose the correct option and choose the correct option.
List I
(A) Maximum value of sin4x + cos4x is (p) 3
List II
(B) Maximum value of is 1 + 8sin2xcos2x is (q) 4
(C) Minimum value of cos4x – sin4x is (r) 1
(D) Minimum value of 4cos2x + 5sin2x is (s) –1
(A) (B) (C) (D)
(1) r s p q
(2) p q r s
(3) r p s q
(4) p q s r
173. Match the items of List I with items of List II and choose the correct option.
List I List II
(A) The minimum value of 2sin2q+3 cos2q is (p) 1
(B) The maximum value of sin2q+ cos4q is (q) 2
(C) The least value of 4[sin4q+ cos2q] is (r) 3
(D) The greatest value of 4[sin2014q+ cos2010q] (s) 4
(A) (B) (C) (D)
(1) q p r s
(2) p q s r
(3) q p s r
(4) p q r s
174. Match the items of List I with the items of List II and choose the correct option.
BRAIN TEASERS
List I List II
(A) If maximum and minimum values of () 2 2 76tantan 1tan θθ θ +− + for all real values of q≠(2 n +1) 2 π are λ and m respectively then (p) λ + m=2
(B) If maximum and minimum values of 5cos + 3cos( q + 3 π )+3 for all real values of q are λ and m respectively then (q) λ – m=6
(C) If maximum and minimum values of 1 + sin( 4 π + q ) + 2cos( 4 π – q ) for all real values of q are λ and m respectively, then (r) λ + m=6 (s) λ – m=10 (t) λ – m=14
(A) (B) (C)
(1) r,s p,q t
(2) r,s r,t p,q
(3) p q,r s,t
(4) p,q r p,s
1. If 44 sincos1 abab θθ += + , then 88 33 sincos ab θθ += (1) ()3 1 ab + (2) ()2 1 ab +
(3) a + b (4) 1 ab +
2. If a sin2x + b cos2x = c, b sin2 y + a cos2 y = d, and atanx = btany, then 2 2 a b =
(1) () () () () adca bcdb (2) () () () () adca bcdb ++ ++
(3) () () () () adba accb (4) () () () () daca bcdb
3. If both q and φ are acute angles and θ=φ=11 sin,cos, 23 then the value of θ + ϕ belongs to
(1) ππ
, 32 (2)
(3) ππ
25 , 36 (4)
2 , 23
5 , 6
4. The sum of the series, sin q sec(3 q ) + sin3 q sec(32q ) + sin(32q ) sec(33q ) + ...... up to n terms, is
(1) θ−θ 1 1 tan3tan3 2 nn
(2) [tan3nq – tan q ]
(3) θ−θ 1 tan3tan 2 n
(4) () θ− 1 tan31 2 n
5. The value of ()() = ∑ 100 1 sincos101 k kxkx is equal to
(1) () 101 sin101 2 x (2) 99 sin(101x) (3) 50 sin(101x) (4) 100 sin(101x)
6. If ππ +=+
tantan3 4242 yx , then + = + 2 2 3sin 13sin x x
(1) sin sin x y (2) cos cos x y
(3) cos cos y x (4) sin sin y x
7. 248 sinsinsin 777 πππ ++= (1) 7 2 (2) 7 2
(3) 7 2 (4) 7 2
8. cosec48° + cosec96° + cosec192° + cosec384° = (1) –2 (2) –1
(3) 0 (4) 3 2
9. 7 coscoscos1 848 ππ−π ++−=
(1) 35 4coscoscos 1648 πππ
(2) 5 4coscossin 1688 πππ
(3) 39 4coscoscos 16816 πππ
(4) 5 4coscoscos 16816 πππ
10. Let () 2 2 2 1tanxtanx Pxcotx 1cotxcotx ++ =+ ++ () 2 cosxcos3xsin3xsinx 2sin2xcos2x −+− + . Then, which of the following is correct?
(1) P(18°) + P(72°) = 2
(2) P(18°) + P(72°) = 3
(3) 47 PP3 36 ππ +=
(4) 47 PP2 36 ππ +=
FLASHBACK (P revious JEE Q uestions )
JEE Main
1. Let the set of all a ∈R such that the equation cos2x+asinx = 2a–7 has a solution [ p, q ] and , 1 tan9tan27tan81 cot63 r =−−+ Then, pqr is equal to _____. (27th Jan 2024 Shift 1)
2. For b ∈ (0, 2 π ), let 3sin( a + b ) = 2sin( a – b ) and a real number k be such that tan a = k tan b . Then, the value of k is equal to ___. (30th Jan 2024 Shift 2) (1) 2 3 (2) –5 (3) 2 3 (4)5
3. If () 2 2 1 tan ,tan 1 1 x AB xxxxx == ++++ and () 1 321 2 tan ,0,,, 2 CxxxABC π =++<< then A+B is equal to ___. (1 st Feb 2024 Shift 1) (1) C (2) π – C (3) 2 π –C (4) 2 π –C
4. The value of tan9°–tan27°–tan63°+tan81° is ______. (6th Apr 2023 Shift 2)
5. The value of 36(4cos29°–1)(4cos227°–1) (4cos281°–1)(4cos2243°–1) is (8th Apr 2023 Shift 2) (1) 18 (2) 36 (3) 54 (4) 27
6. 24816 96coscoscoscoscos 3333333333 πππππ is equal to (10th Apr 2023 Shift 1) (1) 4 (2) 3 (3) 1 (4) 2
7. If the line x = y = z intersects the line xsinA+y sinB+zsinC – 18 = 0 = x sin2A+y
sin2B+zsin2C–9, where A, B, and C are the angles of a triangle ABC, then 80sinsinsin 222 ABC is equal to ____. (15th Apr 2023 Shift 1)
8. If f : R → R is a function defined by ()(){} log2sincos2fxmxxm =−+− for some m , such that the range of f is [0, 2], then the value of m is ____.
(25th Jan 2023 Shift 2) (1) 5 (2) 3 (3) 4 (4) 2
9. If 11 tan15 tan1952 tan75tan105 a +++= , then the value of 1 a a + is (30th Jan 2023 Shift 1) (1) 4 (2) 2 (3) 3 53 2 (4) 423
10. 3579 2sinsinsinsinsin 2222222222
is equal to (25th Jul 2022 Shift 2) (1) 3 16 (2) 1 16 (3) 1 32 (4) 9 32
11. The value of 2sin(12°)–sin(72°) is: (25th Jul 2022 Shift 2) (1) ()513 4 (2) 15 8 (3) ()315 2 (4) ()315 4
12. 16sin(20°)sin(40°)sin(80°) is equal to: (26th Jul 2022 Shift 2) (1) 3 (2) 23 (3) 3 (4) 43
13. Let
Then, (27 th Jul 2022 Shift 2)
(1) 12
14. The value of
is equal to: (27th Jun 2022 Shift 1)
(1) 1 7 and IVth quadrant
(2) 7 and Ist quadrant
(3) –7 and IVth quadrant
(4) 1 7 and Ist quadrant
17. If sin2(10°)sin(20°)sin(40°)sin(50°)sin(70°) = α–1 16 sin(10°), then 16+ α–1 is equal to___
(26th Jun 2022 Shift 1)
JEE
Advanced
18. Let α and β be real numbers such that 0 44 ππ βα −<<<< . If sin(α+β)
= 1 3 and cos(α–β) = 2 3 , then the greatest integer less than or equal to 2 9sincoscossin 4cossinsincos αβαβ βαβα +++ is __ (2022 Paper 2)
19. For non-negative integers n, let
+ + ∑ ∑ 0 2 0 12 sinsin 22 . 1 sin 2 n k n k kk fnnn k n
15. a = sin36° is a root of which of the following equations? (27th Jun 2022 Shift 2)
(1) 10x4–10x2–5 = 0
(2) 16x4+20x2–5 = 0
(3) 16x4–20x2+5 = 0
(4) 16x4–10x2+5 = 0
16. If cotα = 1 and 5 sec 3 β =− where
παβπ <<<< 3 and 22 , then the value of tan(α+β) and the quadrant in which α+β lies, respectively are: (28th Jun 2022 Shift 2)
Assuming cos –1 x takes value in [0, π ], which of the following options is/are correct? (2019 Paper2)
(1) () 3 4 2 f =
(2) () 1 lim 2 n fn ∞→ =
(3) If α = tan(cos–1f(6)), then α2+2α–1 = 0
(4) sin(7cos–1f(5)) = 0
CHAPTER TEST – JEE MAIN
Section - A
1. 23456 cotcotcotcotcotcot 161616161616 7 cot is 16 ππππππ π (1) 0 (2) 1 (3) 1 2 (4) 2
2. If 2 31 ,then 2cot 4 sin π απα α <<+ is equal to
(1) 1 + cotα (2) –1 – cotα (3) 1 – cotα (4) –1 + cotα
3. If sin q + cos q = a, then sin4q + cos4q =
(1) ()2 2 1 11 2 a +− (2) ()2 2 1 11 2 a ++ (3) ()2 2 1 11 2 a (4) ()2 2 1 11 2 a −+
4. If () 1 sincos kk k fxxx k =+ , k = 1, 2, .., then xR∀∈ the value of f4(x) – f6(x) is (1) 1 5 (2) 1 12 (3) 1 12 (4) 5 12
5. If cos x + cos y + cos z = 0 = sin x + sin y + sinz, then tan(x – y) = (1) 2 (2) 3 (3) 1 3 (4) 1 3
6. In a triangle PQR, ∠=π R 2 . If
tan and tan 22 PQ are the roots of ax2 + bx + c = 0, a ≠ 0 then (1) a = b + c (2) c = a + b (3) b = c (4) b = a + c
7. If oo oo 11 tan15tan1952, tan75tan105 a +++= then the value of + 1 a a is: (1) 4 (2) 423 (3) 2 (4) 3 53 2
8. If 22 cos coscos 33 y xz θππ
, then x + y + z is equal to (1) 1 (2) 0 (3) –1 (4) 2
9. 2sin2b + 4cos( a + b )sin a sin b + cos2( a + b ) = (1) sin2 a (2) cos2 a (3) tan2 a (4) cot2 a
10. cos4a + sin4a – 6sin2a cos2a = (1) cos2 a (2) sin2 a (3) cos4 a (4) sin4 a
11. sin3 q .cos3q + cos3 q .sin3q = (1) 3sin4 q (2) 3 sin 4 2 θ
(3) 3 sin 4 4 θ (4) 2sin4 q
12. tan2 a – tan a (1 + sec2 a ) = (1) sin a (2) cos a (3) 0 (4) tan a
13. If, in a triangle ABC, cos3A + cos3B + cos3C = 1, then one angle must be exactly equal to (1) 3 π (2) 2 3 π (3) π (4) 4 3 π
14. cos( a + b + γ ) + cos( a – b – γ ) + cos( b – γ – a ) + cos( γ – a – b ) =
(1) 2cos a cos b cos γ (2) 3cos a cos b cos γ (3) 4cos a cos b cos γ (4) 6cos a cos b cos γ
15. If A+C=2B, then cosCcosA sinAsinC = (1) cotB (2) cot2B (3) tan2B (4) tanB
16. sin70cos40 cos70sin40 + = + (1) 3 (2) 3 (3) 1 3 (4) 1/2
17. Period of sinx sin(120° – x)sin(120° + x) is (1) 2 3 π (2) 3 π (3) π (4) 2 π
18. The period of ()() sincos !1!fxxx nn ππ =− + is
(1) (n + 1)! (2) n! (3) 2((n + 1)!) (4) (n – 1)!
19. The minimum and maximum values of cos32sin6 4 xx π +++
are (1) 11, 1 (2) 6, 5 (3) 5, 6 (4) 1, 11
20. If 0 ≤ x < π/3, then the range of () secsec 66 fxxx ππ =−++ is
CHAPTER TEST – JEE ADVANCED 2022 P1 Model
Section – A [Numerical Value Questions]
1. If A + B + C = π and sin2sin2sin2 sinsinsin ABC ABC ++ = ++ sinsinsin 222 ABC λ
, then the value of λ must be _______.
(1) 4 , 3 ∞ (2) 4 , 3 ∞
(3) 4 0, 3 (4) 4 0, 3
Section – B
21. If 10sin4α + 15cos4α = 6 and the value of 9cosec4α + 4sec4α is S, then the value of 25 S is equal to______.
22. If 44 sincos1 549 xx += , then the units digit of the product of digits of 66 64125 cossinxx + is ___.
23. If 2tanA + cotA = tanB, then cotA + 2tan(A – B) = _____.
24. If π α=α∈π
13 cot,, 22 and −π
5 sec,, 32 , then the value of tan( a + b ) is _______.
25. If sin 2 (10°)sin20° sin40° sin50° sin70° sin10 16 ° =α− then 16 + a–1 is equal to ____.
=+
2. The positive integer value of n > 3 satisfying the equation 111 23 sinsinsin nnn
is _______.
3. If 4sin27° = αβ then the sum of the digits in (α + β – αβ + 2)4 is_______.
4. If sinθ + sin3θ + sin5θ +.... + sin(2 n – 1)θ = sin2 sin n λθ θ , then the value of λ is______.
5. The maximum value of 66 1 sincos y xx = + is_____.
6. Period of 32 tan2sec5sin
is _________.
7. The period of function f(x) = {x} + tan2πx + |sin3πx|:({.} denotes fractional part of ) is_____.
8. If costhen32sinsin35 422 AA A
Section – B
[Multiple Option Correct MCQs]
9. If A > 0, B > 0 and A + B = 3 π , then which the following is not the maximum value of tanAtanB ?
(1) 1 3 (2) 1 3 (3) 1 2 (4) 3
10. Let y = sin2x + cos4x. Then, for all real x, (1) the maximum value of y is 2
(2) the minimum value of y is 3 4
(3) y ≤ 1
(4) y ≥ 1 4
11. If the equation sinx(sinx + cosx) = k has real solutions, then k may lie in the interval
(1) 21 0, 2 +
(2) 23,23 −+
(3) 0,23
(4) 1212 , 22 −+
12. coscossinsin sinsincoscos nnABAB ABAB ++ +
(n even or odd) is equal to (1) 2tan 2 nAB
(2) 2cot 2 nAB
(3) 0 (4) 2tan 2 nAB +
13. If 3sinβ = sin(2 α + β), then tan( α + β) –2tanα is (1) independent of α (2) independent of β (3) dependent on both α and β (4) independent of both α and β
14. The value of the expression 248 tan2tan4tan8cot 7777 ππππ +++ is equal to (1) ππ + 22 coseccot 77
(2) tancot 1414 ππ (3) 2 sin 7 2 1cos 7 π π (4) 2 1coscos 77 2 sinsin 77 ππ ππ ++ +
Section – C
[Matrix Matching Questions]
15. Let α and β be the solutions of the equation 3cos2θ + 4sin2θ = 5. Then match the following and chosee the correct answer from the options.
List I List II
(A) tanα + tan β (p) 0
(B) tan(α + β) (q) 4 3
(C) tan(α – β) (r) 1 4
(D) tanα.tan β (s) 1
(A) (B) (C) (D)
(1) s q p r
(2) q s r p
(3) r q p s
(4) r p q s
16. Match the fundamental period of the following functions. Choose the correct answer from the options given below:
List I List II
(A) 1 sincosxx + (p) 2 π
(B) |sinx + cosx| (q) π
(C) |sinx + cosx|+ |sinx –cosx| (r) 3 2 π
(D) sin sin coscos xx xx + (s) 2 π
(A) (B) (C) (D)
(1) s q p r
(2) q s r p
(3) p q p s
(4) r p q s
17. Match the items of List I with items of List II and choose the correct answer from the options given below.
List I List II
(A) The value of + 2 o o 4sec20 cos20 ec is (p) 1
(B) The minimum value of 1cos28sin2 2sin2 xx x ++
() ,0,/2, x ∈π (q) 2
(C) The value of 000 0 8sin40.sin50.tan10 cos80 (r) 3
(D) If cos5sin5 cossin AA AA + = a + bcos4A, then a2/b is (s) 4
(A) (B) (C) (D)
(1) s q p r
(2) q s r p
(3) p q p s
(4) r q s p
18. If cosα + cosβ = 1 2 and sin α + sinβ = 1 , 3 then match the items of List I with the items of List II. Choose the correct answer from the options.
List I List II
(A) cos 2 αβ + (p) 13 12 ±
(B) cos 2 αβ (q) 2 3
ANSWER KEY
Questions
JEE Advanced Level
1,4 (30) 1,2,3,4 (31) 1,3 (32) 2,4 (33) 1,3 (34) 4 (35) 3,4 (36) 1,2 (37) 1,2,4 (38) 1,3,4 (39) 1,2 (40) 1,2 (41) 1,2 (42) 1,2,4 (43) 1,2,3,4(44) 2,4 (45) 1,2,4 (46) 1,2,3,4(47)
Brain Teasers
Chapter Test – JEE Main
Chapter Test – JEE Advanced
Chapter Outline
4.1 Introduction to Trigonometric Equations
4.2 Principal Solutions of Trigonometric Equations
4.3 General Solutions of Trigonometric Equations
4.4 Solving Trigonometric Inequalities
Trigonometric equations involve expressions containing trigonometric functions like sine, cosine,tangent, etc., and usually, we aim to find the values of the variables that satisfy these equations. These equations often arise in various fields such as physics, engineering, and mathematics itself. Understanding trigonometric equations is essential in applications like signal processing, wave mechanics, navigation, and many other areas where periodic phenomena are studied.
4.1 INTRODUCTION TO TRIGONOMETRIC EQUATIONS
Trigonometric equations are equations involving trigonometric functions of one or more variables.
Example: sin 2x – sin 4x + sin 6x = 0.
1. A trigonometric identity is satisfied by all values of unknown angles.
2. A trigonometric equation is satisfied by certain finite or infinite values.
3. If the equations are not satisfied by any unknown angle, then those equations are called impossible equations.
TRIGONOMETRIC EQUATIONS
4. The value of an unknown angle which satisfies given trigonometric equation is called a solution.
5. Since the trigonometric functions are periodic the trigonometric equations may have infinite number of solutions.
6. Solutions of trigonometric equations are of two types: (1) principal solution (2) general solutions.
4.2
PRINCIPAL SOLUTIONS OF TRIGONOMETRIC EQUATIONS
The least values of the unknown angle that satisfy the equation are called principal solutions. Principal intervals of the trigonometric functions are given below.
4.2.1 Principal Solutions of Tigonometric Functions
Equation in the form of sin q = k: There exists unique value a in , 22 ππ satisfying sina = k for k ∈ [–1, 1]. This a is called principal value of q or principal solution for the equation sin q = k.The principal solution for the equation 1 sin 2 q = is 6 π .
Equation in the form of cos q = k: There exists unique value a in [0, π] satisfying cos q = k. for k∈[–1, 1]. This a is called principal value of q or principal solution for the equation cos q = k
The principal solution for 3 cos 2 q = is 6 π
: Trigonometric Equations
Equation in the form of tanq = k: There exists unique value a in , 22 ππ
satisfying tana = k for k∈(–∞,∞). This a is called principal value of q or principal solution for the equation tanq = k. The principal solution for 1 tan 3 q = is 6 π
Equation in the form of cot q = k : The principal value of q satisfying the equation cotq
= k for k ∈ (– ∞,∞ ) lies in ,00, 22
The principal solution for cot3 q = is 6 π
Equation in the form of secq = k: The principal value of q satisfying the equation secq = k for k ∈ (– ∞ , –1] ∪[1,∞) lies in 0,, 22 ππ
The principal solution for 2 sec 3 q = is 6 π
Equation in the form of csc q = k : The principal value of q satisfying the equation csc q = k for k ∈ (– ∞ , –1] ∪[1,∞) lies in ,00, 22
The principal solution for csc q = 2 is 6 π
1 Find the principle solution for 2 cos2 q = 1 Sol. 2 1 cos 2 q= q = 45°, 135°
Try yourself:
1. Find the principal solution for 51 cos2, 4 q=+ q∈ [0, 2π].
Ans: 10 πq=
4.2.2 The Principal Solutions for the Simultaneous Equations
If two or more equations are there to be solved, then find the solutions for the individual
equations and then take the intersection of those solution sets to get the solution for the given simultaneous equations.
2 Principal value of θ satisfying both the equations 11 sin; tan 2 3 q=q= Sol. 1 11 sin0; tan0 2 3 6 Q q=>q=> ∴q∈ ∴q=π
Try yourself:
2. Find the principal solution of the equations 1 sin 2 x = and 3 cos 2 x=
Ans: 5 6 x π =
TEST YOURSELF
1. The principal solution of 13 cos 22q= (1) 75° (2) –75° (3) 105° (4) –15°
2. The principal solution of 12 sin 22 is q=+ (1) 0 1 22 2 (2) 0 1 67 2 (3) 72° (4) 9°
3. If cot q – tan q = 2, then principal value of q is (1) 4 π (2) 2 π (3) π 8 (4) 3 4 π
4. The principal solution of tan 32 is (1) π 12 (2) 12 (3) 11 12 π (4) 13 12 π
Answer Key (1) 3 (2) 2 (3) 3 (4) 2
4.3 GENERAL SOLUTIONS OF TRIGONOMETRIC EQUATIONS
The set of all solutions of a trigonometric equation is called solution set or general solution.
The general solution should be given unless the solution is required in a specified interval.
4.3.1 General Solutions of the Equation sinq = k
1. The general solution of the equation s in q = k , k ∈ [–1, 1] is { n π + (–1) n a:
n ∈ Z }, where , 22 ππ a
is the principal solution for the equation
sinq = k. General solution for the equation
sin q = 0 is q = {nπ, n ∈ Z}
2. The general solution of the equation csc q
= k, k ∈ (– ∞ , –1] ∪[1,∞) is {nπ + (–1)na:
n ∈ Z }, where ,00, 22 ππ a
is the principal solution for the equation
csc q = k.
3 Find the solution set of 1025 sin 3 4 q=+ Sol. () () 10256 sin 3 sin 415 6 321 15 2 1 : 315 n nnnZ +πq== q=π+-π ππ q=+-∈
Try yourself:
3. Find the solution for the equation 6 sin2 x –5 sin x –1 = 0 . Ans: )(1: 6 n nnZ π π+-∈
4.3.2 General Solutions of the Equation cosq = k
1. The general solution of the equation cosq = k, k∈[–1, 1] is {2nπ ±a: n∈Z} where a∈[0,π] is the principal solution for the equation cosq = k General solution for the equation cos q = 0 is q = { () 21 2 n π + , n ∈ Z .
2. The general solution of the equation sec q = k, k∈(–∞, –1]∪[1,∞) is {2nπ ±a: n∈Z} where 0,, 22 ππ aπ
is the principal solution for the equation sec q = k
4. Find the general solution of 2 sin2q = 3 cosq.
Sol. () () () 2 2 2 2sin3cos 21cos3cos 2cos3cos20 2cos1cos20 q=q -q=q q+q-= q-q+= 1 cos ,cos2 2 q=q=- not posible coscos 3 2; 3 nnZ q=π π q=π±∈
Try yourself:
4. If 2 cos2 q + cos q – 1 = 0, then find q . Ans: 5 2: 2: 36 nnZnnZ ππ π±∈∪π±∈
4.3.3 General Solutions of the Equation
tanq = k
1. The general solution of the equation tanq = k, k∈R is {nπ +a: n∈Z} where , 22 ππ a ∈-
is the principal solution for the equation
tanq = k. General solution for the equation tan q = 0 is q = {nπ, n ∈ Z}.
2. The general solution of the equation cot q = k , k ∈ R is { nπ +a : n ∈ Z } where ,00, 22 ππ a
is the principal solution for the equation cot q = k.
5. Find the solution set for the () 2 tan13tan30 q-+q+= Sol. ()() () () 2 tantan3tan30 tantan13tan10 tan1tan30 tan1,tan3 ,, 43 nnZnnZ q-q-q+= qq--q-= q-q-= q=q=
Try yourself:
5. Find the general solution of 2 sinsec3tan0 qq+q= . Ans: q = nπ, n ∈ Z
4.3.4 General Solutions for the Equation Involving Square Functions
1. General solutions for the equation sin2q = sin2a is q = {nπ ±a , n ∈ Z} where a is the principal solution for the equation sin q = sin a .
2. General solutions for the equation cos2q = cos2a is q = {nπ ±a , n ∈ Z} where a is the principal solution for the equation cos q = cos a .
3. General solutions for the equation tan2q = tan2a is q = {nπ ±a , n ∈ Z} where a is the principal solution for the equation tan q = tan a. 6
Try yourself:
6. If 11 sin2x + 7 cos2 x = 8 then find x Ans: , 6 nnZ
4.3.5
General Solutions of Two Simultaneous Equations
First, find the value of unknown angle q lying between 0 and 2π, satisfying the two or more given equations separately. Select the angle q which satisfies the both the equations, then general solution is given by q = {2nπ + a , n ∈ Z}, and a is the principal solution of q lying between 0 and 2π.
7. Find most general values of θ satisfying the equation ()()2 2 12 3 tan10 Sin +q+q-= Sol. ()()2 2 3 12sin 3 tan10 12sin0 3 tan10 1 1 sin tan 2 3 sin' & tan' 7
Try yourself:
7. Find the most general value of θ which satisfies both the equations tan q = –1 and 1 cos 2 q=
Ans: 7 2, 4 nnZ π π+∈
4.3.6 General Solution for the Equation a cosq + b sin q = k
We know that the range of the expression a cos q + b sin q is 2222 , abab -++
1. If 2222 , kabab∉-++ then the solution set for the equation a cos q + b sin q= k is empty set.
2. If 2222 , kabab∈-++ then the general solution for the equation a cos q + b sin q= k is 2nπ ±a + b, where a is principal solution for the equation () 22 cos k ab qb-= + , 22 cos , a ab b = + and 22 sin b ab = + b
8. If 3cossin2, q+q= then find q.
Sol. 3cossin2 311 cossin 22 2 coscos 64 /62/4 n q+q= ⇒q+q= ππ ⇒q-= ⇒q-π=π±π
Try yourself:
8. If 3 cos q + 4 sin q = λ has a solution. Then find the range of λ.
Ans: [–5, 5]
4.3.7 Solving Trigonometric Equations Involving sin x and cos x
1. If the equation is homogeneous equation in sinx, cosx then divide the equation by cosx and reduce the equation as product of elementary equations. Hence, it can be solved.
• If the given equation is of the form a cos2x + b sin2 x + c sin x cos x + d = 0 then divide the equation on both sides with cos2x and then solve it.
2. If the equation is in the form of f(sinx –cosx, sinx cosx) = 0, then substitute sinx – cosx=t, reduce the given equation as 2 1 ,0 2 ftt= . Hence can be solved.
3. If the equation is in the form of f (sinx + cosx, sinx cosx) = 0, then substitute sin x + cosx=t, reduce the given equation as 2 1 ,0 2 ftt= . Hence can be solved.
9. If 0 ≤ x ≤ 2π, then find the number of solutions of 3(sinx + cosx) – 2(sin3 x + cos3 x) = 8.
Sol. The given equation is 3(sinx + cosx) – 2(sin3x + cos3x) = 8 or (sin x + cos x)[3–2 sinx cosx] = 8 or (sin x + cos x) [sin2 x + cos2 x + 2 sinx cosx] = 8 or (sin x + cos x) = 8 or sin x + cos x = 2. The above solution is not possible. Hence, the given equation has no solution.
Try yourself:
9. Find the general solution of the equation sincos22sincos0 xxxx +-=
Ans: 1 2: (1): 4 64 nZnxnnZn +
4.3.8 Solving Trigonometric Equations Based on Extreme Values
If the equation is in the form of sin q = 1 x x + , check the range of functions of both sides. The range of sinq is [–1, 1]and the range of 1 x x + is (– ∞ , –2] ∪[2,∞) . Since the range sets are disjoint, the given equation has no solution.
10 Find the solution of
Try yourself:
10. Find the solutions for the equation sin4 x = 1 + tan8 x.
φ
4.3.9 Solving Trigonometric Equations Using Graphs
Consider equation f ( x ) = g ( x ) where f ( x ) is a trigonometric function and g ( x ) is either trigonometric or non-trigonometric function. In many cases, we cannot find the exact values of x which satisfy the equation. To find the number of roots of the equation, we draw the graphs of y = f(x), y = g(x), and then find the number of their point of intersection.
11. Find the interval in which the sm allest positive root of the equation tan x – x = 0 lies.
Try yourself:
11. Find the number of roots of the equation x2 = cot x in [0, 2π].
Ans: 2
4.3.10
Number of Solutions and Particular Solutions
To find the solution in the given interval (a,b):
1. Solve the given trigonometric equation.
2. Write the general solution.
3. Put n = 0, ± 1, ± 2, ...... in the general solution and check whether they belong to the specified interval ( a,b) or not.
4. The above step gives the solution of the trigonometric equation in the interval (a, b).
12. Find the number of solutions of |cosx| = sinx, 0 ≤ x ≤ 4π.
Sol. cos sin 04 cos sin (cos 0) tan 1 4 cos sin (cos 0) 3 tan 1 4
No. of solutions 2in(0,2)
Therefore, The number of solutions: 4 in (0, 4π).
Try yourself:
12. Find the number of roots of the equation, ()()sin22 cos 818130 xx += in the interval [0, π].
Ans: 2
TEST YOURSELF
1. For n ∈ Z, 2sin2q = cos2 q implies q 2 =
(1) 2 3 n (2) n 6
(3) n 212 (4) 2 12 n
2. If tantan tantan , 23 23 then the general solution of q is
(1) 31 3 nnZ :
(2) 31 9 nnZ :
(3) 31 6 nnZ :
(4) 21 9 nnZ :
3. If 4cos q cos(120° + q ) cos(120° – q )= 1 2 then q =
(1) nornnZ 31 9 ,
(2) nnZ n 3 1 6 ,
(3) 2 39 nnZ , (4) nnZ 318 ,
4. The solution set of sin3 10 25 4 is
(1) nnZ n 3 1 6 :
(2) nnZ n 3 1 12 :
(3) nnZ n 3 1 2 15 :
(4) nnZ n 3 1 8 :
5. The general solution of 4102 5 cos
(1) 2 3 10 nnZ :
(2) nnZ n 1 3 10 :
(3) nnZ 10 : (4) 2 5 nnZ :
6. If cos cos 3 22 1 1 2 , then
(1) nnZ 3 :
(2) 2 6 nnZ :
(3) 2 3 nnZ :
(4) nnZ 6 :
7. If the sum of all solutions of the equations 8 66 1 2 1 coscos cos xxx in [0, π] is kπ, then [k] = ______. (where [.] represents greatest integer function).
8. The number of solutions of sin3 x = cos2x in the interval 2 , is ________.
9. If 0 2 x , then the number value of x satisfying sinx – sin2x + sin3x = 0 is ___.
Answer Key
(1) 3 (2) 2 (3) 3 (4) 3
(5) 1 (6) 3 (7) 1 (8) 1
(9) 2
4.4 SOLVING TRIGONOMETRIC INEQUALITIES
To solve the trigonometric inequation of the type f ( x ) ≤ a or f ( x ) ≥ a where f ( x ) is some trigonometric ratio, the following steps should be taken:
1. Draw the graph of y = f(x) in an interval length equal to the fundamental period of f(x).
2. Draw the line y=a.
3. Take the portion of the graph for which the inequation is satisfied.
4. To generalise, add nT and take the union over the set of integers, where T is the period of f(x).
13. If 4 sin2 x – 8 sinx + 3 ≤ 0, 0 ≤ x ≤ 2π, then find x.
Sol. () () 2 4sin8sin 30,02 2sin 3 2sin 10 1 sin ,1 2 5 ,
Try yourself:
13. sFind the solution set of sin x < 1. Ans: 2, 2 RnnZ π -π+∈
TEST
YOURSELF
1. The solution set of sin x 1 2 is (1) 2 6 2 6 nn , (2) 2 6 2 7 6 nn , (3) 2 4 2 7 4 nn , (4) 2 6 2 5 6 nn ,
2. The solution set of cos x > 0 is (1) 2 2 2 2 nnnZ ,, (2) 2 2 2 2 nnnZ ,, (3) nnnZ ,, 1 (4) nnnZ 3 1 ,,
3. If A = {x ∈ [0, 2 π ]/ tan x – tan2 x > 0} and Bxx 02 1 2 ,/ sin , then A ∩B is (1) 0 6 , (2) , 7 6 (3) 0 4 7 6 ,, (4) 0 6 5 6 7 6 ,,
4. If sincos31 then (1) 33 (2) 44 (3) 62 (4) 22
5. All the pairs (x, y) that satisfy the inequality 22 5 4 1 2 2 sinsin sin xx y also satisfy the equation: (1) 2sinx = 2 siny (2) sinx = 2 siny (3) sinx = |siny| (4) 2|sinx| = 3 siny
6. If 0 ≤ x ≤ 2π and 2
yy then number of ordered pairs of (x, y) is __.
Answer Key
(1) 2 (2) 2 (3) 1 (4) 3 (5) 3 (6) 2
Important points to solve trignometric equations:
1. Avoid squaring the equation whenever
CHAPTER REVIEW
Principal, General Solutions of Trigonometric Equations
1. The following table shows the principal solutions and general solutions of simple trigonometric equations.
1
2. General solutions for the equations sin2 q = sin2 a, cos2 q = cos2 a, tan2 q = tan2 a is q = {nπ ±a , n ∈ Z}, where a is the principal solution for the given equation.
possible during the solution of trigonometric equations. If squaring becomes necessary, carefully examine the solution to identify and eliminate extraneous values.
2. Ensure that the denominator does not become zero at any stage while solving the equation.
3. Refrain from cancelling terms that contain unknown variables on both sides of the equation when they are in product form, as doing so may lead to a loss of valid solutions.
4. Verify that the solution set not only satisfies the given equation but also falls within the specified domain of the variable in the equation.
3. Suppose that the given trigonometric equation is in the form of a cos q + b sin q = k
• If 22kab >+ then the above equation has no solution.
• If 22kab ≤+ then the above equation has solution and to find the solution, divide both sides with 22ab + and then reduce the equation in terms of only sine or cosine.
4. If the given equation is of the form a cos2x + b sin2 x + c sin x cos x + d = 0, then divide the equation on both sides with cos2x and then solve it.
5. If the equation is in the form of f (sin x –cos x, sin x cos x) = 0, then substitute sin x – cos x=t, reduce the given equation as 2 1 ,0 2 ftt
. Hence it can be solved.
-
6. If the equation is in the form of f(sinx + cosx, sinx cosx) = 0 then substitute sin x + cos x=t, reduce the given equation as 2 1 ,0 2
Important Instructions
9. For equations of the type sin q = k or cos q = k, is solvable if |k| ≤ 1.
. Hence it can be solved.
7. Consider equation f(x) = g(x), where f(x) is a trigonometric function and g ( x ) is either trigonometric or non-trigonometric function. In many cases, we cannot find the exact values of x which satisfy the equation. To find the number of roots of the equation, we draw the graphs of y = f(x), y = g(x), and then find the number of their point of intersection.
Solving Trigonometric Inequalities
8. To solve the trigonometric inequation of the type f(x) ≤ a or f(x) ≥ a where f(x) is some trigonometric ratio, draw the graph of y = f ( x ) and y=a. Take the portion of the graph for which the inequation is satisfied.
10. Avoid squaring the equations both sides if possible, because it may lead to extraneous solutions.
11. Do not cancel the common factor from the two sides of the equations which are in a product otherwise we may loose some solutions.
12. The answer should not contain such values of q , which make any of terms undefined or infinite.
13. In trigonometric equation if L.H.S. of the given trigonometric equation is always less than or equal to k and RHS is always greater than k, then no solution exists. If both the sides are equal to k for same value of q , then solution exists and if they are equal for different value of q, then solution does not exist.
JEE MAIN LEVEL
Level – I
Principle Solutions of Trigonometric Equations
Single Option Correct MCQs
1. If 2 222 2 cos ,then the value of q is
(1) π 16 (2) π 32 (3) π 64 (4) π 128
2. If cos q – sin q = 1, then q = (1) π (2) 3 2 π (3) 2 3 π (4) π 6
3. If1+cos α +cos 2 α +...=2–20 ,then is
(1) 3 4 π (2) π 4 (3) π 6 (4) π 8
4. The smallest value of q satisfying the equation 3 (tan + cot) = 4 qq is
(1) 2 3 π (2) π 3 (3) π 6 (4) π 12
5. If sin cot cos tan 44 and q is in the first quadrant, then q = (1) π /3 (2) π /2 (3) π /4 (4) π /6
6. If tan q + sec q = 3 , then the principal value of 6 is
(1) π 3 (2) π 4 (3) 2 4 π (4) π 2
General Solutions of Trigonometric Equations
Single Option Correct MCQs
7. If 32 cossin , then (1) nnZ n 1 46 ; (2) 2 46 nnZ ; (3) nnZ 6 ; (4) 2 6 nnZ ;
8. The set of solutions of the equation 31 31 2 sincos is
(1) 2 412 nnZ ; (2) 2 412 nnZ ; (3) 21 412 nnZ n ; (4) nnZ n 1 412 ;
9. If sectan , 12 1 then (1) 2 3 2 2 nnZnnZ ,; (2) 22 4 nnZnnZ ,; (3) 2 5 2 6 nnZnnZ ,; (4) 2 7 2 3 nnZnnZ ,;
y: Trigonometric Equations
10. The set of values of x for which tantan tantan 32 13 2 1 xx xx is
(1) φ (2) { π ⁄ 4}
(3) nnz 4 ; (4) 2 4 nnz ;
11. If tanθ . tan(120° – θ)tan(120° + θ) = 1 3 , then θ =
(1) nnZ 312 :
(2) nnZ 318 :
(3) nnZ 318 :
(4) nnZ 312 :
12. If sin θ sin(60° – θ)sin(60° + θ) = 0.25, then θ =
(1) nπ + (–1)n π/6: n ∈ Z
(2) 2n π ± π/3: n ∈ Z
(3) 2n π/3 + π/6: n ∈ Z
(4) n π/3 ± π/6: n ∈ Z
13. The general solution of x satisfying sin2x. secx + 3 tanx = 0 is given by
(1) x = nπ 2 : n ∈ Z (2) x = nπ 3 : n ∈ Z
(3) x = nπ: n ∈ Z (4) x = nπ: n ∈ Z
14. xRxx :cos cos 22 2 2
(1) 2 3 n , nZ
(2) nn 3 , Z
(3) nn 6 , Z
(4) 2 6 nnZ ,
15. If 1 + cos2θ = 3sinθ cosθ, then θ =
(1) nπ + π 4 , nπ – tan–1 1 2 ; n ∈ Z
(2) nπ –π 4 , nπ – tan–1 2; n ∈ Z
(3) nπ + π 4 , nπ + tan–1 2; n ∈ Z
(4) nπ –π 4 , nπ + tan–1 1 2 ; n ∈ Z
16. sin x = 2sin x cos 2x is satisfied. If x belongs to (1) n nnZ 22 ,
(2) n nnZ 33 ,
(3) n nnZ 46 ,
(4) nnnZ 6 ,
17. The general solution of sec x cos 5x + 1 = 0 is (n ∈ Z)
(1) (2n + 1) π 2 , n π
(2) 2n π± π 6
(3) (2n + 1) π 6 ,(2n – 1) π 4
(4) (2n + 1) π 12 ,(2n – 1) π 8
18. Number of solutions of |cos x| = sinx,
0 ≤ x ≤ 4π, is (1) 8 (2) 4 (3) 2 (4) 6
19. The general solution x for the equation, 9cosx – 2.3cosx + 1 = 0 is
(1) n π: n ∈ Z
(2) nπ 2 : n ∈ Z
(3) 2n π: n ∈ Z
(4) () 21 2 n : n ∈ Z
20. If sin42x + cos42x = sin 2x cos 2x, then x =
(1) (2n + 1) 4 π : n ∈ Z
(2) (4n + 1) 2 π : n ∈ Z
(3) (4n + 1) 8 π : n ∈ Z
(4) (2n + 1) 8 π : n ∈ Z
21. The number of solutions of the equation tan x tan 4x = 1 for 0 < x < π is (1) 5 (2) 1 (3) 2 (4) 8
22. The general solution of cos 2 x – 2tan x + 2 = 0 is
(1) (2n + 1) 3 π , n ∈ Z
(2) (n + 1) 3 π , n ∈ Z
(3) n π + 3 π , n ∈ Z
(4) n π + 4 π , n ∈ Z
23. The equation 3 sinx + cosx = 4 has (1) Only one solution (2) Two Solution (3) Infinitely many solutions
(4) No solution
24. The equation a sin x + cos2 x = 2 a – 7 possesses a solution if (1) a > 6 (2) 2 ≤ a ≤ 6 (3) a > 2 (4) a ≥ 7
25. 4 cosθ – 3 secθ = tanθ, then θ = (1) θ = n π + (–1) n β; n ∈ Z where sin 117 8
(2) θ = nπ + (–1)n β; n ∈ Z where sin 117 8
(3) θ = nπ + (–1)n β; n ∈ Z where
sin 117 4
(4) θ = nπ + (–1)n β; n ∈ Z where
sin 117 4
26. If the solutions of cos pθ + cos qθ = 0 are in AP, then the common difference of AP is
(1) pqpq or (2) 22 pqpq or (3) 22 () ()pqpq or (4) 2 2 pqpq or ()
Numerical Value Questions
27. The number of solutions of sec x cos 5x + 1 = 0 in the interval [0,2 π ] is ___.
28. Number of solutions of the equation tan q + sec q = 2 cos q in [0,2 π ] is ________
29. The number of solutions of sin ex x x 1 is _____.
30. The number of solutions satisfying the equation 11 2 2 4 sinsin sin xxx in [0, 4π] equals____.
31. Total number of solutions of cotcot sin ,, xx x x 1 03 is _____.
32. If α is a root of 25 cos2q + 5 cos q – 12 = 0, 2 , then 25|sin 2 α | is equal to ___.
33. If sincos 22 2 nn n , where n ∈ z, then the number of value of n between 4 and 8 for which the equation is true is ______
34. The number of values of θ in the interval , 22 ππ satisfying the equation () 2 sec 3 q = tan4θ + 2tan2θ is _______.
35. The number of solutions of the equation 2cosx = |sinx| in [–2π, 2π] is _____.
Solving Trigonometric Inequalities
Single Option Correct MCQs
36. The solution set of x in [0, π], for which 2sin2x – 3sinx + 1 ≥ 0 is
(1) 5 0, , 626 πππ π
(2) , 63 ππ
(3) 5 , 6 π π
(4) 2 3 π
37. The set of all x in , 22 ππ
|4sinx – 1|< 5 is given by
(1) 3 , 1010 ππ
-
(3) 3 , 105
satisfying
(2) , 105 ππ-
(4) 3 , 510
38. The values of θ ∈ (0, 2π) for which 2sin2θ – 5sinθ + 2 > 0, are
(1) 5 0,,2 66 ππ π
(2) 5 , 86 ππ
(3) 5 0,, 866 πππ
(4) 41 , 48 π
39. The solution set of sin x < 1 is (1) R (2) , 2 RnnZ π π
(3) 2, 2 RnnZ π π
(4) , 2 RnnZ π
40. The number of values of x ∈ [0, 4π] satisfying 3cossin2 xx-≥ is (1) 2 (2) 0 (3) 4 (4) 8
Level – II
General Solutions of Trigonometric Equations
Single Option Correct MCQs
1. If sin q , 1, and cos2 q are in GP, then the general solution of q is
(1) nnZ n 1 2 ; (2) nnZ n 1 2 1 ;
(3) nnZ n 1 6 ;
(4) nnZ n 1 6 1 ;
2. The general solution of tan q+ tan 2q + tan 3 q = 0 is (1) nnZ/,6only
(2) nnZ,, tan where 1 2
(3) Both 1 and 2
(4) nnZ,, where tan 1 5
3. 1 6 sin, cos, tan qqq are in GP, then q is equal to (n ∈ Z)
(1) 2 3 n (2) 2 6 n
(3) n n 1 3 (4) n 3
4. The general solution of the equation sin x – 3sin 2x + sin3 x = cos x – 3cos2 x + cos 3x is ___ (n ∈ Z)
(1) n 8 (2) n 28
(3) 1 28 nn (4) 2 2 3 1 n cos
5. The value of coscos coscos sincos cos yxyx yxx 22 2 s sin 2 y is 0, if
(1) x = 0
(2) y = 0
(3) x = y
(4) xnynZ 4
6. One solution of the equation 4cos2θ sinθ – 2sin2θ = 3sinθ is
(1) θ = nπ + (–1)n 3 10 : n ∈ Z
(2) θ = nπ + (–1)n 3 10 : n ∈ Z
(3) θ = 2nπ ± π 6 : n ∈ Z
(4) None
7. The number of solutions of 67 0 2 cossin cos xxx
(1) 1 (2) 0 (3) 3 (4) 4
8. If cosx xx sins2in 3 2 1 21 then x =
(1) {nπ + (–1)n π 6 : n ∈ Z} ∪ {nπ: n ∈ Z}
(2) {(2n + 1) π 6 : n ∈ Z}
(3) {n π±π 4 : n ∈ Z}
(4) {2n π±π 8 : n ∈ Z}
9. Common roots of the equations of cos 2 x + sin 2x = cot x and 2cos2 x + cos2 2x = 1
(1) 2n π : n ∈ Z
(2) n π : n ∈ Z
(3) () 21 4 n : n ∈ Z
(4) () 21 2 n : n ∈ Z
10. General solution of (sin sin) cos 35 0 2 xxx
(1) (2n + 1) π –π 6 , n ∈ Z
(2) 2n + π 6 , n ∈ Z
(3) 2n π± 5 6 π , n ∈ Z
(4) n π± π 4 , n ∈ Z
11. If 32tan8 θ = 2cos2α – 3cosα and 3cos2θ = 1, then the general value of α is
(1) 2n π, n ∈ Z
(2) n π± 3 π n ∈ Z
(3) 2n π± 3 π , n ∈ Z
(4) 2n π± 2 3 π , n ∈ Z
12. The least difference between the roots of 4cos x(2–3sin2 x) + (cos 2x+1)= 0; 0 ≤ x≤ 2 π
(1) 2 π (2) 3 π (3) 6 π (4) 4 π
13. For n ∈ Z , the general solution of the trigonometric equation
sin 3 cos 4 sin 2 xxx -+
43cos2sin 33cos 30xxx -+-= (1) 28 nππ + (2) 2 6 nππ + (3) 26 nππ ± (4) 2 6 n π π±
Numerical Value Questions
14. If 0 ≤ x ≤ 2π, then the number of real values of x, which satisfy the equation cos x + cos2 x + cos3 x + cos 4 x = 0, is ______.
15. The number of solutions of the equation sin A – sin2 A = cos A – cos2 A in (0, π ) is _____.
16. The number of values of q satisfying 4 cos q + 3 sin q=5 as well as 3 cos q + 4 sin q = 5 is ________ .
17. If the sum of the roots of the equation cos4x +6 = 7cos2x in the interval [0, 314] is k π , k ∈ R, then (k – 4948) is _____.
18. The number of solutions of the equation coscos 2 3 8 3 1 x in the interval [0, 10 π ] is __ .
19. The number of solutions of the trignometric equation 1 – cos x. cos 5x = sin2 x in [0, 2π] is ______.
20. The number of common roots of simultaneous equations cos2x + sin2x = cotx and 2cos2 x+ cos22x = 1, x ∈ [– π , π ] is ______.
21. The number of solutions of sin2x cos2 x = 1 + cos2 x sin4 x in the interval [0, 2π ] is ______.
22. The number of solutions of the equation cos6 x + tan 2 x + cos6 x tan 2 x = 1, in the interval [0,2 π ] is _______
23. The sum of solutions of sin π x + cos π x = 0 in [0,100] is _____.
24. Number of solutions of the equation sinx = [x], where [ ] denotes greatest integer function, is _____.
Solving Trigonometric Inequalities
Single Option Correct MCQs
25. In which of the following sets, the inequality sin6x + cos6x > 5 8 does not hold good?
(1) , 88 ππ -
(3) 3 , 44 ππ
(2)
35 , 88 ππ
(4) 79 , 88 ππ
26. If x,y ∈ [0,15], then the number of solutions (x,y) of the equation 2 cosecx12 34421 yy -×-+≤ is (1) 13 (2) 17 (3) 15 (4) 5
27. The complete set of values of ,, 2 xx π π ∈
satisfying the inequality cos2 x > |sinx| is
(1) , 66 ππ
(2) 5 ,, 2666
(3) 5 ,, 266 πππ π
(4) 5 ,, 666 πππ π
Multiple Concept Questions
Single Option Correct MCQs
28. Solution set of the equation 33 33sincos33sincos1 xxxx ++= , is
(1) ()1, 66 n xnnI ππ π =+--∈
(2) ()21, 6 xnnI π =+∈
(3) 2, 3 xnnI π π =+∈
(4) ()1, 3 n xnnI π π =+-∈
Numerical Value Questions
29. Number of solution of the equation sinx = [x] (where [.] denotes the greatest integer function) is ________.
30. The number of solutions of equation 8[x2 –x ] + 4[x] = 13 + 12[sinx],([.] denotes GIF) is ______.
Level – III
Single Option Correct MCQs
1. If acosx + bcos3x ≤ 1 ∀ x ∈ R, then |b| (1) is equal to 1 (2) ≤ 1 (3) ≥ 1 (4) None of these
2. For the equation 1 – 2x – x2 = tan2(x + y) + cot2(x + y), (1) exactly one value of x exists (2) exactly two values of x exist
(3) y = –1 + n π + 4 π , n ∈ Z
(4) y = 1 + n π + 4 π , n ∈ Z
3. If tan3x + 3 > 3tanx + tan2x, then
(1) x ∈ ,,; 3234 nnnn
n ∈ Z
(2) x ∈ ,,; 4244 nnnn
n ∈ Z
(3) x ∈ ,,; 6244 nnnn -+∪-+
n ∈ Z
(4) x ∈ ,,; 3244 nnnn -+∪-+
n ∈ Z
4. The number of ordered 5 – tuple (u,v,w,x, y) where u,v,w,x,y ∈ [1,11] which satisfy the inequality 22222 sin3cossincoscos 235uvwxy ++ ⋅⋅
≥720, is_____.
(1) 432 (2) 430 (3) 340 (4) 240
5. Let [x] denote the largest integer ≤ x. If the number of solutions of 2 2 sin4cos 1 xx xx xx +-
= -+
, the value of tan2 kx
is k, then for , 43 x ππ
(1) is 1
(2) lies between 21 and 23 (3) 0
(4) lie between 3 11 and 22
6. The equation sin4x – (k + 2)sin2x – (k + 3) = 0 possesses a solution, if (1) k > –3 (2) k < –2 (3) –3 ≤ k ≤ –2 (4) k ∈ Z
Numerical Value Questions
7. Number of integral values of ' a ' satisfying the equation [cosx]2 + cosx = 2a is (where [ ] denotes GIF)
8. Number of solutions of sin{ x} = cos{x} in [0, 2π] is .(Here { x } is fractional part of x ).
9. If 0 ≤ a ≤ 3, 0 ≤ b ≤ 3, and the equation x2 + 4 + 3cos( ax + b ) = 2 x , has at least one solution, then the value of [a + b] is _____ ([.] denotes GIF).
10. The number of solutions satisfying the equation 23 1coscoscos··· 24 xxx ++++∞ = ,(– π< x <π) is _______.
12. If tanA = 1 2 and tanB = 1 7 , principal value of (2A – B) = k π then k – 1 is _____.
13. Number of solution of sin x x = 10 is ______
14. If 3 2 3 19 9 2 sincosxxxx then sum of values of x is _______.
________.
11. If cosx + cosy + cosz=0 and sinx + siny + sinz = 0 then 2 8 cos 2 xy =
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. If , 22 ππ a ∈- , then find the general solution for sin q = sin a.
2. If a∈ [0, π] then find the general solution for cos q = cos a.
3. What is the general solution for cot q = 0?
4. What is the general solution for sin q = 1?
5. Find the general solution for the equation cos q = 1.
Statement Type Questions
Each question has two statements: statement I (S-I)and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct,
(2) if both statement I and statement II are incorrect,
(3) if statement I is correct, but statement II is incorrect,
(4) if statement I is incorrect, but statement II is correct.
6. S-I : The general solution for the equation tan q = k is n π + a, where a is principal solution for the equation tan q = k.
15. The number of values of α in [0, 2 π ] for which 2sin 3 α – 7 sin 2α + 7sinα = 2, is ________.
S-II : The principle solution for the equation tan q = k is a means a is solution for the equation tan q = k and , 22 ππ a
7. S-I : If k ∈ (–1, 1) then the number of solutions for the equation csc q = k is zero.
S-II : If k ∈ (–1, 1), then the number of solutions to the equation is infinite.
8. S-I : Number of solutions to the equation cos x tan x = 1 in [0, π] is zero.
S-II : Number of solutions to the equation cos x = 1 in [0, π] is 1.
9. S-I : The general solution for the equation sin2 q = sin2 a is nπ ±a.
S-II : The general solution for the equation tan2 q = tan2 a is nπ ±a.
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A),
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A),
(3) if (A) is true but (R) is false, (4) if both (A) and (R) are false.
10. (A) : The general solution for the equation 1 sin 2 q = is () 1 6 n n qππ =+- .
(R) : The principal solution for the equation 1 sin 2 q = is 6 qπ =
11. (A) : The general solution for the equation sec2 q = is 2 4 n qππ =±
(R) : The principal solution for the equation 1 cos 2 x = is 4 qπ = .
12. (A) : One of the principal solution for the equation sin2x – 2sinx cosx – 3 cos2x = 0 is 4 π
JEE ADVANCED LEVEL
Multiple Option Correct MCQs
1. what is the value of x in 0, 2 π
satisfying 3131 42 sincosxx -+ += ?
(1) 12 π (2) 5 12
2. The solution of the equation sec4θ – sec2θ =2 is
(1) ()21: 2 nnZ π +∈
(2) () 5 21:nnZ π +∈
(3) () 3 21:nnZ π +∈
(4) () 1 2 0 1: nnZ π +∈
3. The general solution of the equation cos2x + cos22x + cos23x = 1 is x =
(1) ()21: 2 nnZ π +∈
(R) : 4 π is not a principal solution for the equation tan2x – tanx – 3 = 0.
13. (A) : The solution set of the inequality 3 sin 2 q > in [0, 2π] is 2 , 33 ππ
(R) : The sine function is increasing function.
14. (A) : The principal solution for the equation tan2tan 1 1tan2tan xx xx= + is 4 x π = .
(R) : tan2tan tan(2) 1tan2tan xx xx xx + -= ++
(2) () 3 21:nnZ π +∈
(3) () 4 21:nnZ π +∈
(4) () 6 21:nnZ π +∈
4. The solution(s) of the equation
9cos12x + cos22x + 1 = 6cos6xcos2x + 6cos6x – 2cos2x is/are
(1) , 2 xnnl π π =+∈
(2) 1 4 2 cos, 3 xnnl π =±∈
(3) 1 2 cos, 3 xnnl π=±∈
(4) x = n π , n ∈ l
5. If sin3θ + sinθcosθ + cos3θ = 1, then θ = (n ∈ z)
(1) 2n π (2) 2n π+ 2 π
(3) 2n π –2 π (4) n π
6. Which of the following set of values of x satisfies the equation
()() 22 2sin3sin122sin3sin 229 xxxx -+-+ += ?
(1) x = n π± 6 π , n ∈ I
(2) x = n π± 3 π , n ∈ I
(3) x = n π , n ∈ I
(4) x = 2n π+ 2 π , n ∈ I
7. If ()() 33 cos3sin3 cossin m aqaq qq == , then
(1) 42 2 298 cos2 mm m a -+ =
(2) 2 2 cos m m a=
(3) 22 2 298 cos2 mm m a ++ =
(4) 2 2 cos m m a + =
8. The values of x, between 0 and 2π, satisfying the equation cos3x + cos2x = 3 sinsin 22 xx + are (1) 7 π
9. The expression cos3θ + sin3θ + (2sin2θ – 3) (sinθ – cosθ) is positive for all θ in
(1) 3 2,2, 44 nnnI ππ ππ -+∈
(2) 2,2, 26 nnnI ππ ππ -+∈
(3) 2,2, 33 nnnI ππ ππ
(4) 3 2,2, 44 nnnI ππ ππ -+∈
10. If 7cos2x + sinxcosx – 3 = 0 then x =
(1) x = n π + 3 4 π ; n ∈ Z
(2) x = k π + tan–1 4 3
; k ∈ Z
(3) x = k + tan–1 4 3
; k ∈ Z
(4) x = n π + 3 π ; n ∈ Z
11. If sinx + cosx = 1 y y + for x ∈ [0, π], then
(1) x = 4 π (2) y = 0
(3) y = 1 (4) x = 3 4 π
12. The number of solutions of the equation
8sinx = 31 cossinxx +
(1) 6 if x ∈ (0, 2 π )
(2) 4 if x ∈ (0, π )
(3) 5, if x ∈ 3 0, 2 π
(4) no solution exists
13. Number of solutions of equation cosx + cosy + cosz = –3
(1) one if x,y,z, ∈ [0, π ]
(2) one, if x,y,z, ∈ [0, 2 π ]
(3) two, if x,y,z, ∈ [0, 2 π ]
(4) zero, if x,y,z, ∈ [0, π ]
14. If x + y = 4 π and tanx + tany = 1, then (n ∈ Z)
(1) sinx = 0 always
(2) when x = n π + 4 π , then y = –n π
(3) when x = n π –4 π , then y = n π
(4) when x = n π + 4 π , then y = n π –4 π
15. The number of solutions in [0, π] of the equation sin6x + cos6x = a is
(1) none, if a < 1 4
(2) three, if a = 1
(3) two, if a = 1 4
(4) three, if a = 5 4
16. If 2 2 1 cos cos x x + (1 + tan22y)(3 + sin3z) = 4 (wherever defined), then
(1) x can be a multiple of π
(2) x cannot be an even multiple of π
(3) z can be a multiple of π
(4) y can be a multiple of 2 π
17. If cos(x + 3 π ) + cosx = a has real solutions, then
(1) number of integral values of a is 3
(2) sum of integral values of a is 0
(3) when a = 1, number of solutions for x ∈ [0, 2π] is 3
(4) when a = 1, number of solutions for x ∈ [0, 2π] is 2
18. For the smallest positive values of x and y the equation, 2(sinx + siny) – 2cos(x – y) = 3 has a solution. Then, which of the following is/are true
(1) sin1 2 xy + =
(2) 1 cos 22 xy =
(3) Number of ordered pairs ( x,y) is 2.
(4) Number of ordered pairs ( x,y) is 3.
Numerical/Integer Value Questions
19. The number of distinct solutions of the equation 5 4 cos22x + cos4x + sin4x + cos6x + sin6x = 2 in the interval [0, 2π] is ____.
20. The number of solutions of f '(x) = 0 in the interval 0,2, π if f(x) = xcosx2 – sinx2 , is ____.
21. If α, β, γ, and δ are solutions of the equation tan 4 qπ + =3tan3θ, no two of which have equal tangents. Then 1111 tantantantan abgδ +++= ____.
22. The number of solutions of tan x + secx = 2cosx in [0, 2 π ], is ____.
23. The number of real solutions of the equation sin(ex)cos(ex ) = 2x–2 + 2–x–2, is ____.
24. If the number of ordered pairs (x,y) where x,y ∈ [0, 10] satisfyies () 2 sec 2 1 sinsin.21 2 y xx -+≤ , is 2K, then K is ____.
25. The number of solutions of the equation |x| = cosx is ___.
26. Number of orders pairs ( x,y ) satisfying cosx.cosy = 1, where –π ≤ x ≤ π, –π ≤ y ≤ π, is ___.
27. The total number of solutions of sin4x + cos4x = sinxcosx in [0, 2π] is equal to___.
28. The sum of all distinct solutions of the equation 3 secx + cosecx + 2(tanx – cotx) = 0 in the set (),;0, 2 sxx π ππ =∈-≠± is equal to _____.
29. The number of solutions of the pair of equations 2sin2θ – cos2θ = 0 and 2cos2θ –3sinθ = 0 in the interval [0, 2π] is________.
30. The number of roots of the equation 2sin2θ + 3sinθ + 1 = 0 in (0, 2π) is_____.
y: Trigonometric Equations
31. The number of roots of (1 + tanθ)(1 + sin2θ) = 1 + tanθ for θ ∈ [0, 2π] is______.
32. The total number of solutions of tanx + cotx = 2cosec x in [–2π, 2π] is ______.
Passage-based Questions
(Q.33 – 34)
Consider the equation (cos x – sinx) 1 2tan cos x x +
+ 2 = 0.
33. Number of solutions in (0, 4 π ) is_____.
34. Number of solutions in (0, 40 π ) is _____.
(Q. 35 – 36)
α is a root of the equation (2sin x – cos x ) (1 + cosx) = sin2x, β is a root of the equation 3cos2x – 10cosx + 3 = 0, and g is a root of the equation 1 – sin2x = cosx – sinx: 0 ≤ α, β, γ
35. If the value of sin(α – β) is equal to 126 3kthen K = _____.
36. If cosα + cosβ + cosγ = 332 6 m + , then integral value of m is______.
(Q. 37 – 38)
Consider the system of equations xcos3y + 3xcosy sin2y = 14 --- (1) and xsin3y + 3xcos2ysiny = 13 ---- (2)
37. The number of values of y ∈ [0, 6π] is____
38. The value of sin2y + 2cos2y = ____.
(Q.39 – 41)
Consider the cubic equation x3 –(1 + cosθ + sinθ)x2 +(cosθsinθ + cosθ + sinθ)x – sinθcosθ = 0, whose roots are x1, x2 and x3.
39. Number of values of θ in [0, 2π], for which at least two roots are equal, is ____.
40. Greatest possible difference between two of the roots, if θ ∈ [0, 2 π ], is ______.
(Q. 41 – 42)
Consider the system of equations sinxcos2y = (a2 – 1)2 + 1, cosxsin2y = a + 1.
41. The number of values of x ∈ [0, 2π], when the system has a solution for permissible values of a, is ______.
42. The number of values of y ∈ [0, 2π], when the system has a solution for permissible values of a, is ______.
(Q. 43 – 44)
Let () 22 cos sin 2cos21 cos3sintan bxbx xxxx + =, b ∈ R
43. Equation has solutions, if (1) 11 ,1,0, 23 b
(2) () 1 ,11,0, 3 b
(3) 1 1,0, 3 bR
(4) 1 , 2 b
44. For any value of b for which the equation has a solution, the number of solutions when x ∈ (0, 2 π ) is always (1) infinite
(2) depends upon value of b (3) four (4) two
(Q. 45 – 46)
Whenever the terms on the two sides of the equation are different in nature, then the equations are known non standard forms of an ordinary equation. They cannot be solved by standard procedure. non standard problems require high degree of logic; they also require the use of graphs, inverse properties of functions, and inequalities.
45. The number of solutions of the equation 2cos33 2 xxx =+ is _____.
(1) 1 (2) 2 (3) 3 (4) 4
46. The number of real solutions of the equation sin(ex ) = 5x + 5–x is ______.
(1) 0 (2) 1
(3) 2 (4) infinitely many
(Q.47 – 49)
While solving certain trigonometric equations AM ≥ GM is useful. consider the equation
ab
cos4α + 4sin4β + 2 = 4 2 cosαsinβ, 0,,0,. 22
Which is/are correct
47. The value of tan2α + tan2β is equal to ____.
48. The value of sin2α + sin2β is equal to ____.
Matrix Matching Questions
49. Match the items of List I with the items of List II and choose the correct option.
List - I List - II
(A) tan2q=1 (p) n π± 6 π
(B) cos2q= 1 4 (q) n π± 4 π
(C) sin2q= 1 4 (r) n π± 3 π
(D) cosec2q=1 (s) n π± 2 π (t) n π± 8 π
(A) (B) (C) (D)
(1) q s p r
(2) r p t q
(3) q r p s
(4) s p q r
50. For 0 ≤ x ≤ 2π, match the equations in List I to number of solutions in List II.
List - I
List - II
(A) tan2x + cot2x = 2 (p) 2
(B) sin2x – cosx = 1 4 (q) 0
(C) 4sin2x + 6Cos2x = 10 (r) 1
(D) sinx = 1 (s) 4
Choose the correct answer from the options given below:
(A) (B) (C) (D)
(1) s p q r
(2) s p r q
(3) s q p r
(4) s r p q
51. Match the items of List I with the items of List II and choose the correct option.
List - I List - II
(A) If max{5sin q + 3sin ( q – α)}=7, q∈ R then the set of possible values of α is (p) x = 2n π+ 4 π , n ∈ Z
(B) 2 n x π ≠ and 2 sin3sin2 (cos)1 xx x -+ = (q) x = 2n π± 3 π , n ∈ Z
(C) 1 4 (sin)2cos0 xx+= (r) 2n π+ cos-1 1 3 , n ∈ Z
(D) log5tanx = (log54) (log4(3sinx)) (s) No solution
(A) (B) (C) (D)
(1) q s p r
(2) p s q r
(3) q p s r
(4) r s p q
52. Match the items of List I with the items of List II and choose the correct option.
List - I
(A) If 2 sintan 3 sintan qq φφ ==
54. Match the items of List I with the items of List II and choose the correct option.
List - I
List - II
then the values of q and φ are (p) Infinite number of solutions
(B) The number of solutions of 24 24 sincos 1 cossin xx xx + = + is (q) q = n π± 3 π , φ= n π± 6 π
(C) Solution of cot2 q = cot2 q – tan2 q is (r) n π± 4 π
(A) (B) (C)
(1) p q r
(2) q p r
(3) r q p
(4) p r q
53. Match the items of List I with the items of List II and choose the correct option.
List
- I
List
- II
(A) The general solution of sec2x= (1–tan2x) (p) (2n+1) 2 π
(B) If sin6x=1+cos43x then x= (q) x=n π± 3 π
(C) The general solution of sin3α = 4sinαsin(x+α) sin(x–α) is (r) n π± 8 π
(A) (B) (C)
(1) r p q
(2) p q r
(3) q r p
(4) p r q
List - II
(A) x 3+ x 2+4 x +2sin x =0 in x ∈ [0,2 π ] (p) 4
(B) sin(ex)cos(ex) = 2x–2+2–x–2 (q) 1
(C) sin2x + cos4x = 2 (r) 2
(D) 30|sinx| = x when x ∈ [0, 2π] (s) 0
(A) (B) (C) (D)
(1) p q r s
(2) q r s p
(3) q r r p
(4) q p r s
55. Match the items of List I with the items of List II and choose the correct option.
List - I List - II
(A) The set of all real values of parameter ''p'' for which the equation p cosx–2sinx= 2 + 2 ppossesses at least one real root is (p) [–3,–2]
(B) The set of all real values of parameter 'a' for which the equation cos2x+asinx=2a–7 possesses at least one real root is (q) 31 , 22
(C) The set of all real values of parameter 'a' for which the equation sin4x + cos4x + sin2x + a = 0 possesses at least one real root is (r) [2,6]
(D) The set of all real values of parameter ' a', for which the equation cos4x–(a+b) cos2x–(a+3)=0, is (s) [ 5 –1,2]
(A) (B) (C) (D)
(1) s r q p
(2) p q r s
(3) q p s r
(4) r s p q
BRAIN TEASERS
1. Consider the trigonometric equation 1 22 22 63 3 6 3 cotcos cos sec xx xx x cosec 22 3 2 3 33 sin tancot x xx
Which of the following options is/are correct?
(1) Number of solutions of the equation in [0, 6 π ] is 12.
(2) Number of solutions of the equation in [0, 4 π ] is 4.
(3) Sum of all solutions of the equation in 0, 4 π ] is 16 π.
(4) Sum of all solutions of the equation in [0, 4 π ] is 13 π.
2. Let S1 = {x ∈ [0, 2 π ] : 1+ 2cosxcos2xcos5x = cos2x + cos22x + cos25x},
S2 = {x ∈ (0, 2π) : sinx + cosx + tanx + cotx 1 2 22 1 sec} xx cosec S xRxx x x 3 11 2 2 :sin cos, . whereisG.I.F
n(S) denotes the number of elements in S. The Which of the following is
(1) n(S1) = 13
(2) S3 ⊂ S2
(3) S1∆ S2 = S1∆ S3(A ∆ B = (A ∪ B) – (A ∩ B)
(4) n(S2∪ S3) = 3
3. If sinθ = a has exactly 3 solutions in 0 7 3 , , then the value of a is
(1) 10 25 4 + (2) 51 4(3) 51 4 + (4) 31 22 +
4. If S denotes the solution set of the equation x x x x 1 4 2 2 22 cossin , then
(Power set of a set A is the set of all subsets of A)
(1) S is finite
(2) the power set of S has exactly 4 elements
(3) S is infinite
(4) S ⊆ I, the set of integers.
5. 2 2 si2212 ncoscossin xx , if
(1) xnnI 21 2 ,
(2) tan,xnI 1 2
(3) tan,xnI 1 2
(4) , 2 n xnI π =∈
FLASHBACK (P revious JEE Q uestions )
JEE Main
1. If 2tan2q -5se q = 1 has exactly 7 solutions in the interval [0, nπ 2 ], for the least value of n ∈ N, then K K K N 2 1 is equal to (27th Jan 2024 Shift 2)
5. If m and n, respectively, are the number of positive and negative values of θ in the interval [–π, π] that satisfy the equation 9 cos2coscos3cos 22 qqqq = then mn is equal to...... (25th Jan 2023 Shift 2)
6. Let 22 1tantan ,:9910 22 Sxxx ππ
and tan2 3 xs x b ∈
, then 2 1 (14) 6 b - is equal to_____. (10th Apr 2023 Shift 2) (1) 64 (2) 32
(3) 8 (4) 16
2. If α , ππ -<a< 22 is the solution of 4cos q +5sin q = 1, then the value of tanα is (29th Jan 2024 Shift 1)
(1) 1010 6(2) 1010 12(3) 1010 12(4)
6 -
3. If 2sin 3 x + sin2 x cos x + 4sin x – 4 = 0 has exactly 3 solutions in the interval 0,, 2 n nN π
then the roots of the equation x 2+ nx +( n –3) = 0 belongs to : (30th Jan 2024 Shift 1) (1) (0, ∞ ) (2) (- ∞ , 0) (3) 1717 , 22 (4) Z
4. The number of solutions of the equation 4sin 2x – 4cos3x + 9 – 4cosx = 0 ; x ∈ [–2 π , 2 π ] is (1st Feb 2024 Shift 2) (1) 1 (2) 3 (3) 2 (4) 0
7. The number of elements in the set S = {θ ∈ [0, 2π]: 3cos4θ – 5cos2θ – 2sin6θ + 2 = 0} (11th Apr 2023 Shift 1) (1) 8 (2) 10 (3) 12 (4) 9
8. If the solution of the equation logcosx cotx + 4log sin x tan x = 1, x ∈ 0, 2 π
, is 1 sin 2 -ab +
, where α and β are integers, then α + β is equal to (30th Jan 2023 Shift 1)
(1) 4 (2) 6 (3) 5 (4) 3
9. The set of all values of λ, for which the equation cos22x – 2sin4x – 2cos2x = λ has a real solution x, is (29th Jan 2023 Shift 2)
(1) 3 2, 2
(2) 3 ,1 2
(3) [–2, –1] (4) 1 1, 2
10. The number of solutions of the equation 2 1 coscos cos2
[–3 π , 3 π ] is: (24th Jun 2022 Shift 2) (1) 8 (2) 5 (3) 6 (4) 7
11. The number of solutions of |cos x| = sinx, such that –4π ≤ x ≤ 4π is: (25th Jul 2022 Shift 1) (1) 4 (2) 6 (3) 8 (4) 12
12. If the sum of solutions of the system of equations 2sin2θ – cos2θ = 0 and 2cos2θ + 3sinθ = 0 in the interval [0, 2π] is kπ, then k is equal to____. (26th Jul 2022 Shift 2)
13. Let {} 22 2sin2cos 0,2:8816 S
then
is (26th Jul 2022 Shift 1)
(1) 0 (2) –2 (3) –4 (4) 12
14. Let
15. The number of elements in the set 2 :2cos44 6
is (29th Jul 2022 Shift 2) (1) 1 (2) 3 (3) 0 (4) infinite
16. Let S = {θ ∈ (0, 2π): 7cos2θ – 3sin2θ – 2cos22θ = 2}. Then, the sum of roots of all the equations x 2 – 2(tan2θ + cot2θ) x + 6sin2θ = 0,θ ∈ S is___. (29th Jul 2022 Shift 1)
17. Let 3 , , , , . 22444 S πππππ π
Then the number of elements in the set A={θ ∈ S: tanθ(1 + 5 tan(2θ)) = 5 –tan(2θ)} is___ (28th Jul 2022 Shift 2)
18. The number of elements in the set S = {θ ∈ [–4π, 4π]: 3cos22θ + 6cos2θ – 10cos2θ + 5 = 0} is_____. (29th Jun 2022 Shift 1)
19. The number of solutions of the equation 2θ – cos2θ + 2 = 0 in R is equal to____.
(29th Jun 2022 Shift 1)
20. The number of solutions of the equation sin x = cos2 x in the interval (0, 10) is__.
(29th Jun 2022 Shift 2)
21. The number of values of x in the interval 7 , 44 ππ for which 14cosec2x – 2sin2x = 21 – 4cos2x holds, is_______.
(25th Jun 2022 Shift 1)
22. Match the items of List I with the items of List II and choose the correct option (2022 P1)
List – I List – II
(A) 22 , 33 x ππ
:cosx+sinx=1}
(B) 18 55 , 18 x ππ
: 3 tan3x =1}
(C) 55 66 , x ππ
: 2cos(2x) = 3 }
(D) 44 77 , x ππ
: sinx – cosx=1}
(p) has two elements
(q) has three elements
(r) has four elements
(s) has five elements (t) has six elements
CHAPTER TEST – JEE MAIN
Section – A
1. The least difference between the roots, in the first quadrant (0 ≤ x ≤ 2 π ) of the equation 4cosx(2 – 3sin2x) + (cos2x + 1) = 0, is (1) 6 π (2) 4
(3) 3 π (4) 2
2. Let α and β be any two positive values of x for which 2cosx, |cosx| and 1 – 3cos2x are in G.P. The minimum value of | α – β| is (1) 3 π (2) 6
(A) (B) (C) (D)
(1) p s p s
(2) p p t r
(3) q p t s
(4) q s p r
23. Let f : [0, 2] → R be the function defined by f(x) = (3 – sin(2πx))sin(πx –4 π ) – sin(3πx + 4 π ) If α, β ∈ [0, 2] are such that {x ∈ [0, 2] : f(x) ≥ 0} = [α, β], then the value of β – α is _______. (2020 P1)
24. The number of solutions of the pair of equations 2sin2q –cos2q = 0, 2cos2q –3sinq = 0 in the interval [0, 2 π]is ____ (2007 P1) (1) zero (2) one (3) two (4) four
25. Let a,b,c be three non–zero real numbers such that the equation 3 acosx+2bsinx=c, , 22 x ππ
has two distinct real roots α and β with α+β= 3 π .
Then, the value of b a is______.(2018 P1)
2
3. The general value of ‘θ’ that satisfies the equation tanθtan(120° + θ)tan(120° – θ) = 1 3 is
(1) ()61, 18 nnZ π +∀∈
(2) ()31, 3 nnZ π +∀∈
(3) ()61, 6 nnZ π +∀∈
(4) ()31, 6 nnZ π +∀∈
4. If () () 33 2 sincos cos sincos 1cot qqq qqq-+
–2tanθcotθ = –1, θ ∈ [0, 2π] then
(1) 0, 24 qππ ∈-
(2) 3 , 24 ππqπ
(3) 35 , 24 qπππ
(4) ()0,, 42 qπππ ∈-
5. The general solution of the equation 1sin(1)sin1cos2 1cos2 1sinsin nn n xxx xxx -+…+-+…-
is
(1) (–1)n 3
(2) (–1)n 6 π
(3) (–1)n+1 6 π
+
6. The solution set of the equation cos5x = 1 + sin4x is (1) {n π , n ∈ I} (2) {2n π , n ∈ I} (3) {4n π , n ∈ I} (4) { 2 nπ , n ∈ I
7. The equation
(1) one real solution (2) no solution
(3) more than one real solution (4) two solutions
8. Number of ordered pairs (a,x) satisfying the equation sec2(a + 2)x + a2 – 1 = 0, –π < x <π, is (1) 2 (2) 1 (3) 3 (4) infinite
9. If n is the number of solutions of the equation |cot x | = cot x + 1 sin x (0 < x < 2π), then n = (1) 1 (2) 2 (3) 3 (4) 4
10. The equation cos8x + b cos4x + 1 = 0 will have a solution if b belongs to (1) (–∞, 3] (2) [2, ∞) (3) (–∞, –2] (4) [1, ∞)
11. Given that tanA and tanB are the roots of the equation x 2 – bx + c = 0, the value of sin2(A + B) is (1) ()2 b bc + (2) 2 22 b bc + (3) () 2 22 1 b cb +(4) () 2 2 2 1 b bc +-
12. There is exactly one real x ∈ (0, 2 π ) such that tan2 2 x (cot4x + 1)(cosec2x + tan2x)
= 1. Find the positive integer k such that cos2022x = sinkx.
(1) 2023 (2) 4044 (3) 2696 (4) 1011
13. If 0 ≤x≤ π, 22 sincos 818130 xx+= , then x =
(1) 6 π (2) 4 π (3) 15 π (4) 8 π
14. The number of solutions of the pair of equations 2sin2θ – cos2θ = 0, and 2cos2θ –3sinθ = 0 in the interval [0, 2π] is (1) 0 (2) 1 (3) 2 (4) 4
15. The general solution of the equation
sin x – 3sin2 x + sin3 x = cos x – 3cos2 x + cos3x is ___. (n ∈ Z)
(1) 8 n π π+
(2) 28 nππ +
(3) (1) 28 nnππ -+
(4) 1 2 2cos 3 nπ+
16. If cosθ ≠ 0 and secθ – 1 = (21) - tanθ then θ =
(1) 2n π + 4 π (or) 2n π , n ∈ Z
(2) 2n π + 6 π (or) 2n π , n ∈ Z
(3) 2n π + 8 π , n ∈ Z
(4) 2n π –4 π (or) 2n π , n ∈ Z
17. If 4(sin2x sin4x + sin2x) = 3, then x (1) 2 , 39 nnZ ππ ±∈
(2) , 39 nnZ ππ ±∈
(3) ()1, 39 nnnZ ππ ±-∈
(4) () 2 1, 39 nnnZ ππ ±-∈
18. The solution of 4sin2x +tan2x + cosec2x + cot2x – 6 = 0 is (n ∈ Z).
(1) 4 n π π±
(2) 2 4 n π π±
(3) 3 n π π+
(4) 6 n π π-
19. One root of the equation cosx – x + 1 2 = 0 lies in the interval
(1) 0, 2 π (2) ,0 2 π
(3) , 2 π π
(4) 3 , 2 π π
20. The equation e|sinx| + e–|sinx| + 4k = 0 will have exactly four distinct solutions in [0, 2π] if (1) k ∈ R+ (2) 1 ,0 4 k ∈-
(3) k ∈ [–5, –4] (4) none of these
Section – B
21. The number of distinct real roots of the equation tan22x + 2tan2xtan3x – 1 = 0 in the interval 0, 2 π is ______.
22. Number of ordered pairs (x,y) satisfying sinx + siny = sin(x +y) and |x| + |y| = 1 is _______.
23. Number of solutions of the equation sinsin3sin9 0 cos3cos9cos27 xxx xxx ++= in the interval 0, 4 π is _______.
24. The number of distinct real roots of the equation 2 2 tan 3 1 x xx π =++ is _________.
25. Number of real values of x , satisfying the equation [x]2 – 5[x] + 6 – sinx = 0([.] denotes GIF ) is _______.
CHAPTER TEST – JEE ADVANCED
2022 P1 Model
Section – A
[Integer Value Questions]
1. If the values of 'θ' satisfying sin7θ = sin4θ – sinθ in 0 < θ < 2 π are , m ππ then |l – m|= ____
2. The number of solutions of the equation x + 2tanx = 2 π in the interval [0, 2π] is____.
3. The number of solutions of the equation 22 tansec 323281,0 4 xx x π +=≤≤ , is____.
4. The number of solutions of 5 1 cos5 r rx ∑= in the interval [0, 2π] is______.
5. If x ∈ [0, 2π], then the number of solutions of the equation 3(sinx + cosx) – 2(sin3x + cos3x) = 8 is_________.
6. The total number of solutions of cosx = 1sin2 x - in [0, 2π] is equal to_____.
7. The number of values of θ satisfying the equation sin3θ – sinθ = 4cos2θ – 2, θ ∈ [0, 2π], is______.
8. The number of solutions of sin x + cosx = 1 in [0, π] is______.
Section – B
[Multiple Option Correct MCQs]
9. If sin2x + 1 4 sin2(3x) = sinxsin2(3x), then x is equal to (1) ; 2 n nz π ∈ (2) 2; 3 nnz π π±∈ (3) n π; n ∈ Z (4) ()1; 6 n nnz π π+-∈
10. The values of x , between 0 and 2π, satisfying the equation cos3 x + cos2 x = 3 sinsin 22 xx
, are
11. The value of x satisfying ()() 11 33 2sec1sec11 xx-+-= can be in the interval
(1) , 33 x ππ ∈
(2) , 612 x ππ
∈
(3) 35 , 44 x ππ
(4) , 44 x ππ
12. If sin 2 x – 2sin x – 1 = 0 has exactly four different solutions in x ∈ [0, nπ], then the values of n are (n ∈ N) (1) 5 (2) 3 (3) 4 (4) 6
13. If the equation (cosec2θ – 4)x2 + (cotθ + 3 )
x + cos2 3 2 π = 0 holds true for all real ' x ' then the set of possible value of ' θ ' can be given by (n ∈ Z) (1) 11 2 6 n π π+ (2) 5 2 6 n π π+ (3) 7 2 6 n π π+ (4) 11 6 n π π±
14. The possible solutions of the equation tan2θ + cos2θ = 1 is/are (n ∈ Z)
(1) n π –4 π (2) 2n π + 4 π
(3) n π + 4 π (4) 2n π –4 π
Section – C
[Single Option Correct MCQs]
15. The number of solutions of the equation (sin3x – 1)(| 3 tanx + 1| + |2cos2x – 1|) = 0 in the interval [0, 100π] is (1) 100 (2) 150 (3) 200 (4) 250
16. If the equation x 2 tan 2θ – 2tanθ . x + 1 = 0, 2 11 1log1log bc xx acab
1 10 1log a bc
where a,b,c > 1) have a common root and the 2nd equation has equal roots, then the number of possible values of ' θ ' in [0, 3π] is (1) 2 (2) 4 (3) 8 (4) 3
ANSWER KEY
JEE Main Level
17. Values of x and y satisfying the equation sin7y = |x3 – x
+ sec22y + cos4y are
x
(3) x = 1, y = 2n
, n
2
I (4) x = 2, y = 16 nπ , n ∈ I
18. The number of solutions of the equation 16(sin5x + cos5x) = 11(sinx + cosx) in the interval [0, 2π] is (1) 6 (2) 7 (3) 8 (4) 9
–II
–III
Theory-based Questions
Brain Teasers
Chapter Test – JEE Main
Chapter Test – JEE Advanced (1) 5 (2) 3 (3) 1 (4) 2 (5) 0 (6) 2 (7) 5 (8) 2 (9) 3,4 (10) 1,2,3,4 (11) 1,2,4 (12) 1,3 (13) 1,2 (14) 1,2,3,4 (15) 2 (16) 4 (17) 2 (18) 1
