This outlines topics or learning outcomes students can gain from studying the chapter. It sets a framework for study and a roadmap for learning.
Solved Examples
Specific problems are presented along with their solutions, explaining the application of principles covered in the textbook.
Try yourself:
1.Findthevaluesof a,b if(3a –2, b+3)=(2a–1,3)
Ans: a 1,= b 0=
Solved example
1. If n ( A )=4and n ( B )=3,thenfindthe numberofrelationsdefinedfromtheset A toset B.
Sol.Given: n(A)=4and n(B)=3 Hence,thenumberofrelationsdefined fromtheset A toset B is24×3=212
Try Yourself enables the student to practice the concept learned immediately.
This comprehensive set of questions enables students to assess their learning. It helps them to identify areas for improvement and consolidate their mastery of the topic through active recall and practical application.
TEST YOURSELF
1.If R ={(x,y)/x, y ∈ Z, x2+ y2≤4} isa relationin Z,thenthedomainof R is (1){0,1,2}(2){0,–1,–2} (3){–2,–1,0,1,2}(4){0,1,2,3}
Organised as per the topics covered in the chapter and divided into three levels, this series of questions enables rigorous practice and application of learning.
These questions deepen the understanding of concepts and strengthen the interpretation of theoretical learning.
JEE MAIN LEVEL
LEVEL 1, 2, and 3
Single Option Correct MCQs
Numerical Value Questions
THEORY-BASED QUESTIONS
Single Option Correct MCQs
Statement Type Questions
Assertion and Reason Questions
JEE ADVANCED LEVEL
FLASHBACK
CHAPTER TEST
This comprehensive test is modelled after the JEE exam format to evaluate students’ proficiency across all topics covered, replicating the structure and rigour of the JEE examination. By taking this chapter test, students undergo a final evaluation, identifying their strengths and areas needing improvement.
Level 1 questions test the fundamentals and help fortify the basics of concepts. Level 2 questions are higher in complexity and require deeper understanding of concepts. Level 3 questions perk up the rigour further with more complex and multi-concept questions.
This section contains special question types that focus on in-depth knowledge of concepts, analytical reasoning, and problem-solving skills needed to succeed in JEE Advanced.
Handpicked previous JEE questions familiarise students with the various question types, styles, and recent trends in JEE examinations, enhancing students’ overall preparedness for JEE.
Chapter Outline
4.1 Measurement of Angles
4.2 Trigonometric Ratios and Identites
4.3 Compound Angles
4.4 Multiple and Submultiple Angles
4.5 Transformations
4.6 Periodicity and Extremum Values
Introduction to Trigonometry:
■ Trigonometry studies relationships between angles and sides of triangles.
■ It is widely used in physics, engineering, and computer science.
■ Trigonometric functions help analyze the relationship between angles and sides.
■ Applications include:
Measuring heights and distances.
Analyzing wave behavior in physics.
4.1 MEASUREMENT OF ANGLES
■ An angle XOP is formed by two rays OX and OP, as shown in figure. The point O is called the vertex and half lines are called sides of the angle.
CHAPTER 4
TRIGONOMETRIC FUNCTIONS
■ 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.
Measurement of angles:
■ There are three systems of measurement of angles.
Sexagesimal measure
Centesimal measure
Radian measure
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
of a revolution.
It is represented as 1°.
1° = 60 ' (60 minutes), 1' = 60" (60 seconds)
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.
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.
Try yourself:
■ The relation between systems of measurement of angle is 2 90100 DGR π ==
1Radian180=degree=5717'15'' ° π
1degreeradian=0.0175rad 180 = π
Solved example
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 =
CHAPTER 4: Trigonometric Functions
1. Express 5 3 π in degrees. Ans: 300°
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'
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
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
(1) 1 (2) 1 (3) 1 (4) 2 (5) 1 (6) 24 (7) 26 (8) 5
4.2 TRIGONOMETRIC RATIOS AND IDENTITIES
■ Trigonometric ratios define relationships between angles and sides of triangles.
■ Key ratios: sine (sin), cosine (cos), and tangent (tan).
■ Used in geometric analysis and spatial calculations.
4.2.1 Introduction to Trigonometric Ratios
■ A ray OP makes an angle q with x–axis as shown in the figure. Here, q is acute angle O r Y y P(x,y
sinOppositeside
Hypotensue y r q ==
cosAdjacentside Hypotensue x r q ==
tanOppositeside Adjacentside y x q ==
cotAdjacentside Oppositeside x y q ==
secHypotenuse
Adjacentside r x q ==
cscHypotenuse
Oppositeside r y q ==
■ Among the trigonometric ratios, sin q , csc q are reciprocals to each other, i.e., 1 sin;cos,sec csc = qqq q are reciprocals to each other, i.e., 1 cos sec q q = and tan q , cot q are reciprocals to each other, i.e., 1 tan cot q q =
Trigonometric Identities
■ sin2 x + cos2 x = 1 for all x ∈ R
■ 1 + tan2 x = sec2 x for all ()21, 2 xRnnZ π
■ 1 + cot2 x = csc2 x for all x ∈ R–{nπ, n ∈ Z}
Points to Remember:
■ The identity 1 + tan 2 x = sec 2 x can be written as sec 2 x – tan 2 x= 1. It implies that (sec x – tan x ), (sec x + tan x ) are reciprocals to each other.
■ The identity 1 + cot 2 x = csc 2 x can be written as csc2 x – cot2 x= 1. It implies that (csc x – cot x), (csc x + cot x) are reciprocals to each other.
Solved example
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 2coscos1cos 21cos1sin1sin xxxx xxxx xx xx xxx xxx xxx xxx +++-
2. If 10 sin 4 x + 15 cos 4 x = 6 then find the value of 27 csc 6x + 8 sec 6x.
Ans: 250
4.2.2 Properties of Trigonometric Ratios
■ Transformation of trigonometric ratios in terms of other trigonometric ratios is represented in the following table.
■ Sign convention of trigonometric ratios in each quadrant: The sign of trigonometric ratios of angle q depends on the value of q.
If the angle q lies in the first quadrant
2 qπ
then all the trigonometric ratios are positive.
If the angle q lies in the second quadrant 2
, then sin q , csc q are positive and all other trigonometric ratios are negative.
If the angle q lies in the third quadrant 3 2
then
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
positive and all other trigonometric ratios are negative.
If the angle q lies in the fourth quadrant 3 2 2
then cos q , sec q are positive and all other trigonometric ratios are negative.
■ 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
Increases from – ∞ to 0
Decreases from 0 to – ∞
Increases from – ∞ to –1
Increases from 1 to ∞
Decreases from 0 to –1
Increases from –1 to 0
Increases from 0 to ∞
Decreases from ∞ to 0
Decreases from –1 to – ∞
Increases from – ∞ to –1
Increases from –1 to 0
Increases from 0 to 1
Increases from – ∞ to 0
Decreases from 0 to – ∞
Decreases from ∞ to +1
Decreases from –1 to– ∞
Solved example
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,
1cos tan3 cos x x x
=-=- and
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 -
4.2.3 Graphs of Trigonometric Functions
■ The graph of y = sin x
The domain of y= sin x is R and its range is [–1, 1].
CHAPTER 4: Trigonometric Functions
■ The graph of y = cos x
The domain of y = cos x is R and its range is [–1, 1].
■ The graph of y = tan x y
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 .
■ 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.
■ 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 .
■ 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.
Solved example
4. Draw the graph of 2cos 6 yx
=-
Sol. Given: 2cos 6 yx
The
shifts the graph of y = cosx to 6 π units right.
=-
The transformation () 2cos 6 fxx
stretch vertically the graph of
Therefore the graph of
Try yourself:
4. Draw the graph of
4.2.4 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 33 ,,,,,,, 2222 xxxxxxx --+-+-+ ππππ
and so on.
■ Trigonometric ratios of (– x):
The angle (–x) lies in the fourth quadrant. Cosine and secant functions are positive, remaining all trigonometric ratios are negative
sin(–x) = –sin x, csc(–x) = –csc x
cos(–x) = cos x, sec(–x) = sec x
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.
■ 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
ππ -=-=
ii. cscsec,seccsc 22 xxxx
iii. tancot,cottan 22 xxxx
■ Trigonometric ratios of 2 x π
+
:
The angle 2 x π
+
lies in the second quadrant. In this quadrant, sine and cosecant functions are positive and the remaining trigonometric ratios are negative.
CHAPTER 4: Trigonometric Functions
functions are positive and the remaining trigonometric ratios are negati ve.
sin(π + x ) = –sin x , cos (π + x ) = –cos x
csc(π + x ) = –csc x , sec (π + x ) = –sec x
tan(π + x) = tan x, cot(π + x) = cot x
■ 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.
33 sincos,cossin 22 xxxx ππ -=--=-
33 cscsec,seccsc 22 xxxx
33 tancot,cottan 22 xxxx ππ -=-=
■ Trigonometric ratios of 3 2 x π +
:
sincos,cossin 22 xxxx
ππ +=+=-
cscsec,seccsc 22 xxxx
tancot,cottan 22 xxxx ππ +=-+=-
■ 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.
sin(π – x) = sin x, cos (π – x) = –cos x
csc(π – x) = csc x, sec (π – x) = –sec x
tan(π – x) = –tan x, cot(π – x)= –cotx
■ Trigonometric ratios of (π + x):
The angle (π + x) lies in the third quadrant. In this quadrant tangent and cotangent
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.
33 sincos,cossin 22 xxxx ππ
33 cscsec,seccsc 22 xxxx ππ
33 tancot,cottan 22 xxxx ππ
■ 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.
sin(2π – x) = –sin x, cos(2π – x) = cos x
csc(2π – x) = –csc x, sec(2π – x) = sec x
tan(2π – x) = –tan x, cot(2π – x) = –cot x
Solved example
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
■ If a cos q – b sin q = c then sincos222 ababc qq+=±+-
■ If a cos q + b sin q = a and a sin q≠ 0 then a sin q – b cos q = b
■ If a cos q + b sin q = b and a cos q≠ 0 then b cos q – a sin q = a
■ If a sec q + b tan q = c and a tan q + b sec q = k then a2 – b2 = c2 – k2
■ If a csc q + b cot q = c and a cot q + b csc q = k, then a2 – b2 = c2 – k2
15. If 15 sin4a + 10 cos4a = 6, for some R a ∈ then the value of aa + 2766 sec8cosec is equal to _____.
16. If 1+ 4 tan q = 4 sec q, then 8 17 cosq -=__.
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 __.
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 a+b+γ=π and cot a , cot b , and cot γ are in AP, then cot a , cot γ = ___.
14. If () ()+a+a=a∈
π 1tan1tan42,0, 10 , then 20
a
= ___.
15. sin10° – sin110° + sin130° = ___.
Answer Key
(2)
4.4 MULTIPLE AND SUBMULTIPLE ANGLES
■ Let A be an angle. The measures of angles 2A, 3A, 4A are called multiple angles and the measures of angles ,,,... 234 AAA are called submultiple angles.
4.4.1 Trigonometric Ratios of Multiple and Submultiple Angles
■ For any A ∈ R, the trigonometric ratios of 2A
sin(2A) = 2 sinA cosA
() 2 2tan tan2 1tan A A A = -
Here, A, 2A are not odd multiples of 2 π
() 2 cot1 cot22cot 1 cottan 2 A AA AA
Here, A , 2 A are neither integral multiples nor odd multiples of 2 π
■ Replace A with 2 A in the above relations to get the following.
sin2sincos 22 AA A
22 2 2 cossin 22 cos12sin 2 2cos1 2 AA A A A
=-
2 2cos1cos 2 A A =+
22 coscossin 22 AA A =-
2 2sin1cos 2 A A =
■ 2 2tan tan2 1tan 2 A A A =Here , 2 A A are not odd multiples of 2 π
■ 2 cot1 cot2 2cot 2 A A A=
Here , 2 A A are not odd multiples of 2
CHAPTER 4: Trigonometric Functions
■ 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
■ 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
2 2tan sin2 1tan A A A = +
2 2 1tan cos2 1tan A A A= +
2 2tan tan2 1tan A A A = -
Here 2A is not odd multiples of 2 π
■ Substitute 2 A in place of A in the above formulae
2 2tan 2 sin 1tan 2 A A A
=
+
Here, A is not an odd multiple of π.
2 2 1tan 2 cos 1tan 2 A A A - = +
Here, A is not an odd multiple of π.
2 2tan 2 tan 1tan 2 A A A
= -
Here A is not an odd multiple of , 2 π π
■ Trigonometric ratios of triple angles: For any measure of angle A ∈ R,
sin 3A = 3 sinA – 4 sin3A
cos 3A = 4 cos3A – 3 cosA
3 2 3tantan tan3 13tan AA A A=3A,A are not odd multiples of 2 π
3 2 3cotcot cot3 13cot AA A A=3A,A are not multiples of π
Solved example
9. Find the value of22 22 sin3c3 os sincos AA AA
Sol. LHS:22 22 sin3c3 os sincos AA AA= = = = 2222 22 22 22 22 sin3coscos3sin
RHS: sincos sin2sin4 sincos 2sincos2sin2cos2 sincos 4sincos2sincoscos2 sincos 8cos2 AAAA AA AA AA AAAA AA AAAAA AA A
Try yourself:
9. Find the value of
4.4.2 Expressing the Trigonometric Ratios in Terms of cos 2A
■ We can express all trigonometric ratios in terms of cos 2A.
For all A ∈ R, 1cos2 sin 2 A A=± For all A ∈ R, 1cos2 cos 2 A A + =± For all ()21;, 2 ARnnZ π ∈-+∈
1cos2 tan 1cos2 A A A=± +
For all {} ; ARnnZ π ∈-∈ , 1cos2 cot 1cos2 A A A + =± -
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.
1cos sin 22 AA=± for all A ∈ R
1cos cos 22 AA + =± for all A ∈ R
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.
Solved example
10. If π < x < 2π, then find 1cos 1cos x x + -
Sol. 2 2 2 22 1cos2cos2 1cos2sin 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
4.4.3 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°:
51 sin18 4°=
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
CHAPTER 4: Trigonometric Functions
trigonometric ratios of 1 22 2 °
121 sin22 222°=
121 cos22 222 °=+
1 tan2221 2 °=-
1 cot2221 2 °=+
■ The trigonometric ratios of 36°:
Use the values of sin18°, cos18° to get sin36°, cos36°.
1025 sin36 4°=
51 cos36 4 °=+
■ The trigonometric ratios of 72°:
Use the values of sin18°, cos18° to get sin72°, cos72°
1025 sin72 4 °=+
51 cos72 4°=
11. Find the value sin2 24° – sin2 6°.
Sol. sin(24° + 6°) sin(24° – 6°) sin30° sin 18°
ππ
sinsin 610 15151 248 =×=
Try yourself:
11. Find the value of 22 2 sinsin 55 ππ -
Ans: 5 4
4.4.4 Expressing the Trigonometric Ratios of 2 A in Terms of sin A
■ Use the trigonometric ratios of multiple and submultiple angles formula to get the trigonometric ratios of 2 A in terms of sinA.
()()2cos1sisnn 2 1i A A A =±±++
()()2sin1sisnn 2 1i A A A =±±+-
2 11tan tan 2tan AA A -±+ = Let cos,sin 22 AA CS==
■ If , 244 A ππ ∈- then C+S > 0, C–S > 0
1sin1sin2cos 2 A AA ++-=
1sin1sin2sin 2 A AA +--=
■ If 3 , 244 A ππ ∈
then C+S > 0, C–S < 0
1sin1sin2sin 2 A AA ++-=
1sin1sin2cos 2 A AA +--=
■ If 35 , 244 A ππ ∈
then C+S < 0, C–S < 0
1sin1sin2cos 2 A AA ++-=-
1sin1sin2sin 2 A AA +--=-
■ If 57 , 244 A ππ ∈
then C+S < 0, C–S > 0
1sin1sin2cos 2 A AA ++-= 1sin1sin2sin 2 A AA +--=This can be represented as below.
or 44 ππ -
53 or 44
■ Trigonometric ratios of 1 9,7 2 °° : The values of sin9°, cos9° are as below.
3555 sin9 4 °=+--
3555 cos9 4 °=++-
■ The values of 11 cot7,tan7 22 °° are as below.
1 cot76432 2 °=+++
1 tan76432 2 °=--+
Solved example
12. Find the value of tan 9° – tan 27° – tan 63° + tan 81°.
■ a 2 sin 2 q + b 2 csc 2 q≥2 ab , a 2 cos 2 q + b 2
sec2 q≥2 ab and a2 tan2 q + b2 cot2 q≥2 ab
■ For the function 22222222 cossinsincos, axbxaxbx +++
minimum value is a+b at x= 0 and the
maximum value is ()222 ab + and occurs at 4 x π = .
■ 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 posit ive. 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.
CHAPTER 4: Trigonometric Functions
■ In a triangle ABC, the maximum value of
sin A + sin B + sin C is 33 2
cos A + cos B + cos C is 3 2
sinsinsin 222 ABC is 1/8
coscoscos 222 ABC is 33 8
cos A cos B cos C is 1 8
■ In a triangle ABC, the minimum value of
tan A+ tan B+ tan C is 33
tan A tan B tan C is 33
222 tantantan 222 ABC ++ is 1
2 sinsin1 3 sin AA A ++ ≥
■ If range of f ( x ) is [– a , 0], then range of () 1 fx is 1 , a -∞
Solved example
18. Find the range of 1 5sin12cos13 xx+.
Sol. f(x) = 5 sinx + 12 cosx – 13
Minimum value of f(x) is 22131326cab-+=--=-
Maximum value of f(x) is 2213130cab++=-+=
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
(2) 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
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 sinAB 5 (2) ()+= 1 sinAB 5 (3) () 1 cosA+B 5 = (4) () 1 cosA+B 5=
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 461 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
4: Trigonometric Functions
40. If () sin Asin B3cosBcosA+=- then
Sin3A + Sin3B =
(1) 0 (2) 2222 (3) 1 (4) –1
41. If A + B + C = 720°, then tanA + tan B + tanC =
61. If x 1 and x 2 are two distinct roots of the equation a cos x + b sin x = c , then + tan12 2 xx is equal to (1) a b (2) b a (3) c a (4) a c
Numerical Value Questions
62. If a=b=35 coscos 513 and then 65 a-b = 2 cos 2
63. If ⋅= 1 tantan 2 AB , then (5 – 3cos2A) (5 – 3cos2B) =
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 coscoscoscos 8888 k then 2k = ____
Transformations
Single Option Correct MCQs
68. If 0, 2 AB π << satisfy the equations
CHAPTER 4: Trigonometric Functions
3 sin2A + 2sin2B = 1 and 3sin2A – 2sin2B = 0, then A + 2B =
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 q+q+…+q = q+q+…+q
(1) 0 (2) tan( q 1 + q n)
(3) tan1 2 n q+q (4) tan1 2 n q-q
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 aaa ≤……≤ 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
(3) 3 , 22 ππ
(2) 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 ππ -+ 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
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 222111 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° =
CHAPTER 4: Trigonometric Functions
(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 abγ ⋅⋅
(2) 4coscoscos 222 abγ ⋅⋅
(3) 4sinsinsin 222 abγ ⋅⋅
(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 qqqπ …∈ , 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 161sin 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 -a-b-γ=-a+b+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 () ()() a+b-γγ=b≠γa-b+γb tantan , tantan , 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 coscoscoscos 8888
πππ +++= 444 678 coscoscos 888 ___.
41. If ()=+++ 2 23 sinsin3sin3 cos3cos3cos3 n fxxxx xxx ..... ()() 1 23 sin3,then/4/4 cos3 n n x ff x+π+π= ___.
42.
ππ q-q-
3 16coscoscoscos 88 57 coscoscoscoscos4, 88
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 =a+b , then sum of digits in ( a + b – ab + 2)4 =________.
51. The value of cos61cos62cos119 111 cos1cos2cos59
52. Given that () 2sin2 cos2cos4 fn n q=q q-q
()()()() sin 23, sinsin ffffn q+q+q+…+q=λq qmq 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 = ____.
THEORY-BASED QUESTIONS
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.
1. S-I : The measurement of angle 2 lies in the second quadrant.
S-II : The measure of 1 radian is approximately equal to 57 °.
2. 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.
3. 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
4. S-I : If q lies in the second quadrant, then 2 1sincosqq-=.
S-II : If q lies in second quadrant, cos q is negative.
5. 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
6. 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.
7. 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.
8. S-I : Let 0, 2 qπ ∈ and πqq±=±
sinsin, 2 n when n is even.
S-II : Let 0, 2 qπ
and
CHAPTER 4: Trigonometric Functions
sincos, 2 n when n is odd.
9. 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.
10. 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.
11. S-I : If 2 sectan 3 xx+= , then measure of angle x lies in fourth quadrant.
S-II : If 1 csccot 5 qq+= , then measure of angle q lies in second quadrant.
12. 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 + =.
13. S-I : If tan 15° + cot 195° = 2 a, then the value of 1 4 a a += .
S-II : tan195tan1523 °=°=+
14. 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 + += -
15. S-I : tan 15° + cot 75° = 4
S-II : tan 15° + cot 15° = 4
16. S-I : cos 42° + cos 78° + cos 162° = 0
S-II : cos 162° = cos 18°
17. S-I : If cotcotcot3 ABC++= , then
the triangle ABC is an equilateral triangle.
S-II : If 3,1xyzxyyzzx ++=++= , then x=y=z .
18. 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.
19. 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.
20. S-I : tan50° – tan40° = 2 tan10°
S-II : tan70° – tan20° = 2 tan50°
21. S-I : 4 sin3 A + sin 3A = 3 sin A
S-II : 4 cos3 A – cos 3A = 3 cos A
22. S-I : 1 tan376432 2 °=-+-
S-II : 1 tan526432 2 °=+--
23. 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.
24. S-I : 2 sin 54° sin 66° 1 cos12 2 =°+
S-II : 2 sinA sinB = cos (A–B) – cos (A+B)
25. 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.
26. S-I : The period of sin4 x + cos4 x is right angle.
27. 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,
S-II : The fundamental period of of sin2 x + cos2 x is not defined.
(4) if both (A) and (R) are false.
28. (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.
29. (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.
30. (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.
31. (A) : cos 1 > cos 2
(R) : Cosine function is decreasing function in the first quadrant.
32. (A) : If sin x + csc x = 2, then the value of sinnx + cscnx is 2.
(R) : If 1 2 x x += , then the value of x is 1.
33. (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).
34. (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.
35. (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 +=
36. (A) : The value of () sec1562 °=.
(R) : cos(45° – 15°) = cos 45° sin15° + sin 45° cos15°.
37. (A) : 1 tan2221 2 °=-
(R) : If A+B = 45°, then (1+ tan A)(1 + tan B) = 2
38. (A) : cos11sin11 tan34 cos11sin11 °-° °= °+°
(R) : cossin tan 4cossin AA A AA π-
39. (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.
40. (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.
41. (A) : tan70tan20 1 tan50=
(R) : sin(A–B) = sinA cosB – cosA sinB
42. (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.
43. (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
44. (A) : 223 sinsin1 88 ππ +=
(R) : 223cossin 88 ππ =
45. (A) : sin 3x = sin x(2 cos 2x + 1)
(R) : cos 2x = 2 sin2 x – 1
46. (A) : tan6° tan42° tan66° tan 78° = 1
(R) : tan q tan(60° + q )tan(60° – q ) = tan 3 q
47. (A) : The value of sin20° sin40° sin60° sin
80° is 3 16
(R) : For any value of q, sinq + sin (60° + q) () 1 sin60sin3 4 qq°-= .
(R) : The period of af 1( x ) + bf 2( x ) is the least common multiple of periods of f1(x), f2(x).
53. (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)
54. (A) : The extremum value of a2 sin2x + b2 csc2x is 2ab.
(R) : The extremum value of a2 sec2x + b2 csc2x is (a + b)2.
JEE ADVANCED 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) 1 2
(3) 1 22 (4) 1 2 -
3. If 3 tanA + 4 = 0, then the value of 2cot A – 5cosA + sinA is equal to
(1) π <<π 23if 102 A
(2) π <<π 233if2 102 A
(3) π π<< 53if 102 A
(4) π π<< 533if2 102 A
4. The value of ()aaa =-+ 2 cos2cotfec
aa + 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 q = (2) 4 tan 3 q = (3) () 2 2 tan 1 m m q =(4) () 2 2 tan 1 m m q = +
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 32sin2cos1, 25 yxx y + =++ + then the value of y lies in the interval
(1) 8 , 3 ∞ (2) 12 , 5 ∞
(3) 812 , 35
- (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 a =+ and 11 cos,(0) 2 yxy y b =+> ,,,, xyR ab ∈ , then
(1) () sinsin R abγγ∀γ ++=∈
(2) coscos1;, R ab∀ab=∈
(3) 2 (coscos)4;,R ab∀ab +=∈
(4) sin(α + b + γ ) = sinα + sin b + sin γ ;
12. If 44 sincos1 235 xx += , then
(1) 22 tan 3 x =
(2)
88 sincos1 827125 xx +=
(3) 21 tan 3 x =
(4) 88 sincos2 827125 xx +=
13. If 0 ≤ q ≤ π and 22 sincos 818130 qq+= , 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 2cos223secqq += , where q∈(0, 2π), then which of the following can be correct?
(1) 1 cos 2 q = (2) tan q = 1
(3) 1 sin 2 q =- (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) ()12 2log1 x bb - (2) 2log(1 – bx/2)
(3) log12 x bb - (4) 12 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 ∞∞ =φφ =
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 ∠q = . 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 q +q
(3) 1 1sin+q
(4) 2 1sin cos q q -
22. Which of the following is/are correct?
(1) ()()() lnsinlnsin (tan)(cot),0,/4 xx xxx∀π>∈
(2) () lnln 45,0,/2 cosecxcosecx x ∀π<∈
(3) ()()() lncoslncos (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) ()a= + 2 2 4 tan () abab ab
(4) () + a=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)
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== 312cotandtan 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 a=<a<π 3 sin, 52 and -πb=π<b<53 cos, 132 then the correct statements is/are
(1) ()a-b= 63 tan 16
(2) ()a+b= 33 tan 56
(3)a= 24 sin2 25
(4)b= 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 20 (2) π 30 (3) π 4 (4) π 2
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) +22 122 xy xy (2)+ 22 122 xy xy
(3) + + 22 122 xy xy (4)22 122 xy xy
35. For <q<π 0 2 , the solution (s) of () = -ππ q+
6 1 1 coseccosec42 44 m mm
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) a-b=-π 2 3 (2) γ-b=π
37. If ()()() b-γ+γ-a+a-b=-
coscoscos 2 )() b-γ+γ-a+a-b=- 3 coscoscos 2 , then
(1) ∑ cos a = 0
(2) ∑ sin a = 0 (3) ∑ cos a sin a = 0 (4) ∑ (cos a+ sin a ) = 0
CHAPTER 4: Trigonometric Functions
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 ()q= -q 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) + 2 1 3
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) π
--a 22sin 4 (2) π
(4)
2 2cos 4
2 2cos 4
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) q() =+ 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 qqqqq=+ 7 2sinsin 22 qq ++ ..... + () qq +∈2sinsin21, 22 nnN then which of the following is/are correct?
(1)
(2)
9 1 42 f
2 0, n fnN n
(3) 51 2 f π
(4) 9 51 42 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
94 sec 105 cosec
(4)
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 1cos1cos 44 kk kk . then, (1) ()= 1 3 16 P (2) ()=
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 333 48 xx-=- is satisfied by (1)
52. If
tansin 2 x cosecxx , then
2tan 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) ()22 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 qqq +q++…+q= qqq +q++…+then (1) 3 3 21 16 f π =
(2) 521 28 f π =+
(3) 723 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
62. If xcos a + ysin a = xcos b + ysin b 20, 2 a π =<ab< , then
(1) 22 4 coscos ax xy a+b= +
(2) 22 22 4 coscos ay xyab= +
(3) 22 4 sinsin ay xy a+b= +
(4) 22 22 4 sinsin ax xyab= +
63. If sin a + sin b = l, cos a cos b = m, and ()tantan1 22 n ab
, then
(1) () 222 cos 2 lm a-b=+-
(2) () 22 22 cos ml mla+b= +
(3) 122 12 nlm nn ++ = -
(4) if 2 lm π a+b==
64. Iftansincos,then sincos q=a-a a+a
(1) sincos2sin a-a=±q
(2) sincos2cos a+a=±q
(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) 2229sinsinsin 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 tan2tan0, tan4fxxx 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 12 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 22 nnxy ab
________.
76. If 2tanq = secq, then 4cot2q – 3tan2q =_____.
CHAPTER 4: Trigonometric Functions
77. If p cosec q + q cot q = 2 and p 2 cosec 2 q –q2cot2q = 5, then the value of 8122 pqis ______.
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 223sincos 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 qqq qqq -++++ 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 -<b<ππ <a< 0 44 If ()a+b= 1 sin 3 and ()a-b= 2 cos 3 then the greatest integer less than or equal to 2 9sincoscossin 4cossinsincos abab +++ baba is ___.
88. If A + B + C = π , then the value of ++ coscoscos sinsinsinsinsinsin ABC BCACAB is ___.
89. If a=b= ++ 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 ++ 22223 cotcotcot 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 a= , 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 qqq =++ qqq then 1 2 k k = ___.
97. ()()sin22tan Ifsinthen 5tan ABAB B A ++ == ____.
98. If A + B + C = π and sin2sin2sin2 sinsinsin sinsinsin222 ABCABC ABC ++=λ ++ , then the value of λ must be ___.
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 2222 sinsin 33 ππqq ++
is______.
104. If the period of (() ) cossin , tan nx nN x n ∈ is 6π, then n = ____.
105. If 3cos5cos3 3 ab π qq ≤+++≤ , then the value of () 2 bais
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 q∈ 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 loglog11 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 (2) 2 3 (3) 3 3 (4) 4 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 abqq = and 22 , cossin axbyab qq ==- then (x, y) lies on
CHAPTER 4: Trigonometric Functions
(1) a circle (2) a parabola (3) an ellipse (4) a hyperbola
(Q: 115 – 116)
In ∆ ABC, BC = 1, 123sin,sin,cos 222 ABA xxx === , and cos4 2 B x = with 20072006 13 24 0 xx xx
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 2222222 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) 32 4 a (2) 32 4 b
(3) 32 4 c (4) 1 2 abc
(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) b -b sin 1cos A A (2) b +b sin 1cos A A
(3) b -b cos 1sin A A (4) b +b cos 1cos A A
128. The value of tan b is
(1)
(2)
(3)
(4)
() a+b ab sin1cos coscos A A
() a-b ab sin1cos coscos A A
() a-b ab cos1sin coscos A A
() a+b ab cos1sin coscos A A
129. Which of the following is the value of tan( a + b )?
(1) b bsin sin A (2) aa b-a 2 sincos cossin A
(3) aa b+a 2 sincos cossin A (4) a asin cos A
58
(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 q+=q 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 +++ abγδ 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
If a = sin10°, b = sin50°, c = sin70° then answer the following
139. The value of + + 8 abcab 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 sinsin11 andcoscos 43 a+b=a+b= , 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
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 a+b= and 1 sinsin 2 a+b= 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., 2222 cossin cababccab qq -+≤±+≤++
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 qqπ +++ lies between (1) –4 and 10 (2) 4 and 10 (3) –4 and –1 (4) 4 and –10
156. If 3cos5sin 6 b aqqπ ≤+-≤ 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
CHAPTER 4: Trigonometric Functions
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 qq+= and sincos1,xy ab -=qq then 22 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 qq=<q<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 List II
(A) 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
ππqq (p) () 1 1sin2 2 +q
(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
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 (p) Equilateral triangle
(B) In ∆ ABC, if tanA > 0, tanB > 0 and tanAtanB < 1, then triangle is (q) Right angle triangle
(C) In ∆ ABC, if cos3 A + cos3 B + cos C = 3cosAcosBcosC then triangle is (r) Acute angle triangle
(D) In ∆ ABC cot A > 0, cot B > 0 and cot Acot B < 1, then triangle is
(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 + cos4xis (p) 3
(B) Maximum value of is 1 + 8sin2xcos2x is (q) 4
(C) Minimum value of cos4x– sin4x is (r) 1
List II
(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.
List I List II
(A) If maximum and minimum values of () 2 2 76tantan 1tan qq q ++ for all real values of q≠(2 n +1) 2 π are λ and m respectively then
(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
(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
(p) λ + m=2
(q) λ – m=6
(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
4: Trigonometric Functions
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 ()22 1 tan,tan 11 x AB xxxxx == ++++ and () 1 tan3212,0,,, 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 tan15tan1952 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 πππππ
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 () 9 1 0,:
Then, (27 th Jul 2022 Shift 2) (1) 12 S π
14. The value of 246 coscoscos
is equal to: (27th Jun 2022 Shift 1) (1) –1 (2) 1 2(3) 3 1 - (4) 4 1 -
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 b =- where ππ πabπ <<<< 3and 22 , then the value of tan(α+β) and the quadrant in which α+β lies, respectively are: (28th Jun 2022 Shift 2)
(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 ba -<<<< . If sin(α+β)
= 1 3 and cos(α–β) = 2 3 , then the greatest integer less than or equal to 2 9sincoscossin 4cossinsincos abab baba
is __
(2022 Paper 2)
19. For non-negative integers n, let ()
Assuming cos –1 x takes value in [0, π ], which of the following options is/are correct? (2019 Paper2)
(1) () 3 4 2
(2) () lim1 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
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
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) sinsin 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 2 1cos28sin 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.
5.2 Principal Solutions of Trigonometric Equations
5.3 General Solutions of Trigonometric Equations
5.4 Solving Trigonometric Inequalities
■ Trigonometric equations involve sine, cosine, tangent, etc., to find variable values that satisfy them.
■ These equations appear in physics, engineering, and mathematics.
■ Applications include:
Signal processing
Wave mechanics
Navigation
Other areas involving periodic phenomena
5.1 INTRODUCTION TO TRIGONOMETRIC EQUATIONS
■ Trigonometric equations are equations involving trigonometric functions of one or more variables.
Example: sin 2x – sin 4x + sin 6x = 0.
A trigonometric identity is satisfied by all values of unknown angles.
A trigonometric equation is satisfied by certain finite or infinite values.
CHAPTER 5
TRIGONOMETRIC EQUATIONS
If the equations are not satisfied by any unknown angle, then those equations are called impossible equations.
The value of an unknown angle which satisfies given trigonometric equation is called a solution.
Since the trigonometric functions are periodic the trigonometric equations may have infinite number of solutions.
Solutions of trigonometric equations are of two types: (1) principal solution (2) general solutions.
5.2 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.
5.2.1 Principal Solutions of Tigonometric Functions
■ Equation in the form of sin q = k: There exists unique value a in , 22 ππ satisfying sin a = 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 θ = 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 θ = is 6 π .
■ Equation in the form of tanq = k: There exists unique value a in , 22
satisfying tan a = k for k ∈ (– ∞,∞ ). This a is called principal value of q or principal solution for the equation tan q = k. The principal solution for 1 tan 3 θ = 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 θ = is 6 π
■ Equation in the form of sec q = k : The principal value of q satisfying the equation
sec q = k for k ∈ (– ∞ , –1] ∪[1,∞) lies in 0,, 22
The principal solution for 2 sec 3 θ = is 6
■ 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
Solved example
1. Find the principle solution for 2 cos 2 q = 1
Sol. 21 cos 2 θ= q = 45°, 135°
CHAPTER : Trigonometric Equations
Try yourself:
1. Find the principal solution for 51 cos2, 4 θ=+ q∈ [0, 2π].
Ans: 10 πθ=
5.2.2 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.
Solved example
2. Principal value of θ satisfying both the equations 11 sin;tan 23 θ=θ= Sol. 1 11 sin0;tan0 23 6 Q θ=>θ=>
∴θ∈
∴θ=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 22 θ= (1) 75° (2) –75° (3) 105° (4) –15°
2. The principal solution of 12 sin 22 is θ=+ (1) 10 22 2 (2) 10 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 tan32 is (1) π 12 (2) 12 (3) 11 12 π (4) 13 12 π
5.3 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.
5.3.1 General Solutions of the Equation
sinq = k
■ The general solution of the equation s in q = k , k ∈ [–1, 1] is { n π + (–1) n a:
n ∈ Z }, where , 22 ππ α
is the principal solution for the equation
sinq = k. General solution for the equation
sin q = 0 is q = {nπ, n ∈ Z}.
■ The general solution of the equation csc q = k , k ∈ (– ∞ , –1] ∪[1,∞) is { n π + (–1) n a:
n ∈ Z }, where ,00, 22 ππ α
is the principal solution for the equation
csc q = k.
Solved example
3. Find the solution set of 1025 sin 3 4 θ=+
Sol. () () 10256 sin 3sin 415 6 321 15 2 1: 315 n nnnZ +πθ==
Try yourself:
3. Find the solution for the equation 6 sin2 x –5 sin x –1 = 0 . Ans: )(1: 6 n nnZ π
5.3.2 General Solutions of the Equation cosq = k
■ 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 .
■ The general solution of the equation sec q = k, k ∈ (– ∞ , –1] ∪[1,∞) is {2nπ ±a : n ∈ Z} where 0,, 22 ππ απ
∈∪
is the principal solution for the equation sec q = k.
■ The general solution of the equation tan q = k, k ∈ R is {nπ +a : n ∈ Z} where , 22
is the principal solution for the equation tanq = k General solution for the equation tan q = 0 is q = {nπ, n ∈ Z}.
■ The general solution of the equation cot q = k , k ∈ R is { nπ +a : n ∈ Z } where ,00, 22 ππ α
is the principal solution for the equation cot q = k.
Solved example
5. Find the solution set for the () 2 tan13tan30 θ−+θ+= . Sol. ()() () () 2 tantan3tan30 tantan13tan10 tan1tan30 tan1,tan3 ,, 43 nnZnnZ θ−θ−θ+= θθ−−θ−= θ−θ−=
Try yourself:
5. Find the general solution of 2 sinsec3tan0 θθ+θ= . Ans: q = nπ, n ∈ Z
5.3.4 General Solutions for the Equation Involving Square Functions
■ 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
■ 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 .
■ 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.
Solved example
6. The value of 'x' satisfying 4 sin2 x = 1 is
Try yourself:
6. If 11 sin2x + 7 cos2 x = 8 then find x.
Ans: , 6 nnZ π
5.3.5 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.
Solved example
7. Find most general values of θ satisfying the equation ()() 22 123tan10 Sin +θ+θ−= .
Sol. ()() 22 3 12sin3tan10 12sin03tan10
sin'&tan' 7 66 7 2: 6 vevein nnZ +θ+θ−=
Try yourself:
7. Find the most general value of θ which satisfies both the equations tan q = –1 and 1 cos 2 θ= Ans: 7 2, 4 nnZ π π+∈
5.3.6 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 −++
■ If 2222 , kabab∉−++ then the solution set for the equation a cos q + b sin q= k is empty set.
■ 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 θβ−= + , 22 cos, a ab β = + and 22 sin b ab = + β
8. If 3 cos q + 4 sin q = λ has a solution. Then find the range of λ.
Ans: [–5, 5]
5.3.7 Solving Trigonometric Equations Involving sin x and cos x
■ 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.
■ If the equation is in the form of f (sin x –cosx, sinx cosx) = 0, then substitute sinx – cos x=t , reduce the given equation as 12 ,0 2 ftt
. Hence can be solved.
. Hence can be solved.
■ If the equation is in the form of f (sinx + cosx, sinx cosx) = 0, then substitute sinx + cos x=t , reduce the given equation as 21 ,0 2 ftt
Solved example
9. If 0 ≤ x ≤ 2π, then find the number of solutions of 3(sinx + cosx) – 2(sin3x + cos3x) = 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: 12:(1): 644 nZnxnnZn +
5.3.8 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 sin q 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.
Solved example
10. Find the solution of 222 2 1 2cossin 2 x xx x =+
CHAPTER y: Trigonometric Equations
Try yourself:
10. Find the solutions for the equation sin 4 x = 1 + tan8 x.
Ans: φ
5.3.9 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.
Solved example
11. Find the interval in which the sm allest positive root of the equation tan x – x = 0 lies.
Therefore, the required interval is
Try yourself:
11. Find the number of roots of the equation x2 = cot x in [0, 2π].
Ans: 2
5.3.10 Number of Solutions and Particular Solutions
■ To find the solution in the given interval (a,b):
Solve the given trigonometric equation.
Write the general solution.
Put n = 0, ± 1, ± 2, ...... in the general solution and check whether they belong to the specified interval ( a,b) or not.
The above step gives the solution of the trigonometric equation in the interval (a,b).
Solved example
12. Find the number of solutions of |cosx| = sinx, 0 ≤ x ≤ 4π.
Sol. cossin04 cossin(cos0)
tan1 4 cossin(cos0) 3 tan1 4
No.ofsolutions2in(0,2) xxx xxx xx xxx xx =≤≤π => π =⇒= −=< π =−= π Therefore, The number of solutions: 4 in (0, 4π).
Try yourself:
12. Find the number of roots of the equation, ()() 22 sincos 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 tantantant 23an, 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 1025 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 41co025 s
(1) 2 3 10 nnZ :
(2) nnZ n 1 3 10 : (3) nnZ 10 : (4) 2 5 nnZ :
6. If cos cos 3 221 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 co1 scoscos 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 ___.
CHAPTER y: Trigonometric Equations
5.4 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:
Draw the graph of y = f(x) in an interval length equal to the fundamental period of f(x).
Draw the line y=a.
Take the portion of the graph for which the inequation is satisfied.
To generalise, add nT and take the union over the set of integers, where T is the period of f(x).
Solved example
13. If 4 sin2 x – 8 sinx + 3 ≤ 0, 0 ≤ x ≤ 2π, then find x.
Sol. () () 2 4sin8sin30,02 2sin32sin10 sin1,1 2 5 , 66 xxx xx x
Try yourself:
13. Find the solution set of sin x < 1. Ans: 2, 2 RnnZ
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 sincos 31 then (1) 33 (2) 44 (3) 62 (4) 22
5. All the pairs (x, y) that satisfy the inequality 225 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 1 2 12 co22 s ecx 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:
■ Avoid squaring the equation whenever possible during the solution of trigonometric equations. If squaring becomes necessary, carefully examine the solution to identify and eliminate extraneous values.
■ Ensure that the denominator does not become zero at any stage while solving the equation.
■ 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.
■ Verify that the solution set not only satisfies the given equation but also falls within the specified domain of the variable in the equation.
# Exercises
JEE MAIN
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 =
3. If1+cosα+cos2α+...=2–20 ,then is
4. The smallest value of q satisfying the equation 3(tan+cot)=4 θθ 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
CHAPTER y: Trigonometric Equations
General Solutions of Trigonometric Equations Single Option Correct MCQs
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
(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
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
n ∈ Z
CHAPTER y: Trigonometric Equations
(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 22 sin4cos 1 xx xx xx +− = −+
, the value of 2tan 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 _______.
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 α
2. 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.
3. S-I : Number of solutions to the equation cos x tan x = 1 in [0, π] is zero.
________.
11. If cosx + cosy + cosz=0 and sinx + siny + sinz = 0 then 82 cos 2 xy =
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 si2 ncos xxxx then sum of values of x is _______.
15. The number of values of α in [0, 2 π ] for which 2sin 3 α – 7 sin 2 α + 7sinα = 2, is
THEORY-BASED QUESTIONS
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.
1. 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.
S-II : Number of solutions to the equation cos x = 1 in [0, π] is 1.
4. 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.
5. (A) : The general solution for the equation 1 sin 2 θ = is () 1 6 n n θππ =+− .
(R) : The principal solution for the equation 1 sin 2 θ = is 6 θπ =
6. (A) : The general solution for the equation sec2 θ = is 2 4 n θππ =±
(R) : The principal solution for the equation 1 cos 2 x = is 4 θπ = .
7. (A) : One of the principal solution for the equation sin2x – 2sinx cosx – 3 cos2x = 0 is 4 π
(R) : 4 π is not a principal solution for the equation tan2x – tanx – 3 = 0.
8. (A) : The solution set of the inequality 3 sin 2 θ > in [0, 2π] is 2 , 33 ππ
(R) : The sine function is increasing function.
9. (A) : The principal solution for the equation tan2tan 1 1tan2tan xx xx = + is 4 x π = .
(R) : tan(2)tan2tan 1tan2tan xx xx xx + −= ++
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 π (3) 7 24 π (4) 11 36 π
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 =
CHAPTER y: Trigonometric Equations
(1) ()21: 2 nnZ π +∈
(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) 142 cos, 3 xnnl π =±∈
(3) 12 cos, 3 xnnl π =±∈
(4) x = n π , n ∈ l
5. If sin 3θ + sinθcosθ + cos 3θ = 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 αθαθ θθ == , then
(1) 42 2 298 cos2 mm m α −+ =
(2) 22 cos m m α=
(3) 22 2 298 cos2 mm m α ++ =
(4) 22 cos m m α + =
8. The values of x, between 0 and 2π, satisfying the equation cos3x + cos2x = 3 sinsin 22 xx + are
(1) 7 π (2) 5 7 π (3) 9 7 π (4) 13 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 θπ + =3tan3θ, no two of which have equal tangents. Then 1111 tantantantan αβγδ +++= ____.
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 ____.
CHAPTER y: Trigonometric Equations
24. If the number of ordered pairs (x,y) where x,y ∈ [0, 10] satisfyies () 2 2sec 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_____.
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 ≤ 2 π .
35. If the value of sin(α – β) is equal to 126 3k then 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 cossin 2cos21cos3sintan 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 cos4α + 4sin4β + 2 = 4 2 cosαsinβ, 0,,0,. 22 ππαβ
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
CHAPTER y: Trigonometric Equations
(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 List - II
(A) If 2 sintan 3 sintan θθ
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
54. Match the items of List I with the items of List II and choose the correct option.
List - I
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 p possesses 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
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 12 is equal to (27th Jan 2024 Shift 2) (1)
2. If α , ππ −<α< 22 is the solution of 4cos q +5sin q = 1, then the value of tanα is (29th Jan 2024 Shift 1)
(1)
(3)
6 (2)
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 x2+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 (1 st Feb 2024 Shift 2) (1) 1 (2) 3 (3) 2 (4) 0
CHAPTER y: Trigonometric Equations
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 θθθθ = then mn is equal to...... (25 th Jan 2023 Shift 2)
6. Let 22 1tantan ,:9910 22 Sxxx ππ
and 2 tan 3 xs x β ∈ =
, then 12 (14) 6 β is equal to_____. (10th Apr 2023 Shift 2)
(1) 64 (2) 32
(3) 8 (4) 16
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 sin1 2 αβ + , 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 (29 th 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 12 coscoscos2 334 xxx
[–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)
15. The number of elements in the set 2 :2cos44 6 Sxxxxx
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(tan 2θ + cot 2θ) x + 6sin 2θ = 0,θ ∈ S is___. (29 th Jul 2022 Shift 1)
17. Let 3 ,,,,. 22444 S πππππ
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_____. (29 th Jun 2022 Shift 1)
19. The number of solutions of the equation 2θ – cos2θ + 2 = 0 in R is equal to____.
(29 th Jun 2022 Shift 1)
20. The number of solutions of the equation sin x = cos2 x in the interval (0, 10) is__.
(29 th 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_______.
(25 th 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
(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
CHAPTER y: Trigonometric Equations
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)
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 π (3) 3 π (4) 2 π
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 π (3) 2 π (4) 1 22 cos 33 π
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 sincoscos sincos1cot θθθ θθθ +
–2tanθcotθ = –1, θ ∈ [0, 2π] then
(1) 0, 24 θππ
(2) 3 , 24 ππθπ
(3) 35 , 24 θπππ
(4) ()0,, 42 θπππ ∈−
5. The general solution of the equation 1sin(1)sin1cos2 1sinsin1cos2 nn n xxx xxx −+…+−+…−
is
(1) (–1)n 3
(2) (–1)n 6 π
(3) (–1)n+1 6 π
(4) (–1)n–
+ n 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 cos 4x + 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 22 1 b bc +−
12. There is exactly one real x ∈ (0, 2 π ) such that 2 tan 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) 2cos12 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±
CHAPTER y: Trigonometric Equations
(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 tan3 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 = 4cos 2θ – 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 (1) 7 π (2) 5 7 π (3) 9 7 π (4) 13 7 π
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 2 θ – 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
–II
–III
Theory-based Questions
CHAPTER y: Trigonometric Equations
17. Values of x and y satisfying the equation sin7y = |x3 – x2 – 9x + 9| + |x3 – x2 – 4x + 4| + sec22y + cos4y are (1) x = 1, y = n π , n ∈ I (2) x = 1, y = 2n π+ 2 π , n
I (3) x = 1, y = 2n π , n ∈ 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
Chapter Test – JEE Main
Chapter Test – JEE Advanced
Chapter Outline
INVERSE TRIGONOMETRIC FUNCTIONS
6.1 Domain and Range of Inverse Trigonometric Functions
6.2 Properties of Inverse Trignometric Functions
6.3 Solving Inverse Trigonometric Equations
6.4 Telescopic Series
6.5 Standard Results
■ The inverse of a function f : A → B exists if f is one-one and onto, i.e., a bijection, and it is given by f (x) = y ⇒ f–1 (y) = x
■ f : A → B is a bijective function ⇔ f–1 : B → A exists and is a bijective function. The inverse of a bijective function is unique.
■ Trigonometric functions are not bijective functions, so their inverse does not exist. By restricting the domain and codomain, trigonometric functions can be made invertible. sin–1 x is an angle and denotes the smallest numerical angle, whose sine is x. Similarly, cos–1 and tan–1 can be defined.
6.1 DOMAIN AND RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS
■ The function :,1,1 22 f
, defined by () sin fxx = , is a bijection. Then, 1:1,1, 22 f
ππ is also a bijection. This function is called inverse sine function and it is denoted by sin–1x or arc sin x. sin–1x is different from (sin x)–1. The former is the measure of an angle in radians whose sine is x, while the latter is 1/sin x
■ The function f : [0, p ] → [–1, 1], defined by f ( x ) = cos x , is a bijection. Then, f –1: [–1, 1] → [0, p] is also a bijection. This function is called inverse cosine function and it is denoted by cos–1x or Arc cosx.
■ The function ():,, 22 f ππ −→−∞∞ , defined by () tan fxx = , is a bijection. Then, () 1 :,, 22 f −∞∞→− ππ is also a bijection. This function is called inverse tangent function and it is denoted by tan–1x or Arc tan x.
CHAPTER 6: Inverse Trigonometric Functions
Solved example
1.If 1113 sinsinsin, 2 αβγπ ++= then find the valueof αβαγβγ ++
Sol. We know that, 1 111 sin 22 sin,sin,sin 222 x ππ πππαβγ −≤≤ ∴=== 1.Thus,3αβγαβαγβγ ∴===++=
Try yourself:
1. Find the value of a for which ()()21212sin22cos220axxxxx+−++−+= has a solution.
Answer: 2 π
TEST YOURSELF
1. The domain of the function cos –1(2x – 1) is (1) [0, 1] (2) [−1, 1] (3) (−1, 1) (4) [0, π]
2. The domain and range of f ( x ) = sin –1 x + cos–1x + tan–1x + cot–1x + sec–1x + cosec–1x, respectively, are (1) {} 3 1,1, 2 π (2) {}1,1, 2
π (3) ()1,1, 2 π (4) (–1, 1), {2 p }
3. If the domain of the function f(x) = loge(4x2 + 11x + 6) + sin–1(4x + 3) + 1106 cos 3 x +
is (α, β], then 8|α + β| is equal to (1) 20 (2) 10 (3) 30 (4) 50
5. If sin–1x + sin–1y + sin–1z = 3 2 π , then
201201402402
4. If sin–1x + sin–1y + sin–1z = 3 2 π , the value of x100 + y100 + z100 –101101101 9 xyz ++ is equal to (1) 0 (2) 1 (3) 2 (4) 3
■ Inverse trigonometric functions satisfy the following properties in their respective domains. Properties related to addition, subtraction, and multiple angles of inverse trigonometric functions are discussed below.
6.2.1 Identities for (–x)
■ Inverse trigonometric functions satisfy the following properties for negative values of x.
11sin()sinxx−=− 1,1 x ∀∈−
11 cos()cosxx π −=− 1,1 x ∀∈−
11tan()tanxx−=− xR∀∈
11cot()cotxx π −=− xR∀∈
11 sec()secxx π −=−() 1,1 xR ∀∈−−
1–1 cosec()cosecxx−=− () 1,1 xR ∀∈−−
Solved example
2. What is the range is () 11 cotcot? x yx =
Try yourself:
2. Find the value of 1 cot(3)is Answer: 5 6
6.2.2 Identities for Reciprocals of (x)
■ Inverse trigonometric functions satisfy the following properties for reciprocal values of x. 11 1 sincosec x x
CHAPTER 6: Inverse Trigonometric Functions
Solved example
3. Find the value of 111tantan x x + . Sol. 1 1 1 11 11 11 1cot;0 tan cot;0 1 tantan tancot,0 2 tancot;0 2 xx
Try yourself:
3. What is the value of the sum 111111 cos2sin3cos 222
Answer: 31 12 π
6.2.3 Identities of the Form f(f–1 (x))
■ Inverse trigonometric functions satisfy the following properties of composite function of trignometric functions and their inverse.
sin(sin–1 x) = x, if –1 ≤ x ≤ 1
1 cos(cos), xx = if 11 x −≤≤
tan 1 (tan), xx = if x −∞<<∞
1 cot(cot), xx = if x −∞<<∞
1 sec(sec), xx = if 1 x −∞<≤− or 1 x ≤<∞
cosec–1(cosec), xx = if 1 x −∞<≤− or 1 x ≤<∞
Solved example
4. Simplify ()() 2121 sectan2coseccot3. +
Sol. Given:
Try yourself:
4. What is the value of () 1 cotcot2024 ?
Answer: 2024
6.2.4 Principal Value of Inverse Trigonometric Functions of the Form f–1 (f(x))
■ Inverse trigonometric functions in the form of f –1(f(x)) can be expressed in principal values, as shown below.
()
Examples:
sin–1(sin 1) = 1, sin–1(sin 2) = π – 2,
sin–1 (sin 3) = π – 3, sin–1(sin 4) = π – 4, sin–1(sin 5) = 5 – 2π, sin–1 (sin 10) = 3π – 10. sin–1(sinx) is a periodic function with period
cos –1 (cos x ) is a periodic function with period 2π . ■ () 1 3 , 22 , tantan22 3 , 22 35 2, 22 xx xx yx
Examples:
=2p+x y= x
tan–1(tan 1) = 1, tan–1(tan 2) = 2 – π, tan–1 (tan 3) = 3 – π, tan–1(tan 4) = 4 – π, tan–1(tan 5) = 5 – 2π, tan–1 (tan 10) = 10–3π, tan–1(tanx) is periodic function with period p . y= x–p y = x–2p y = x+p
■ () 1 2,2 ,0 cotcot,0 ,2 2,23 xx xx yxxx xx xx πππ
() 1 cotcot x is periodic function with period π
■ () 1 coseccosec ,,00, 22 3 ,,, 22 yx
■ () 1 secsec ,0,, 22 33 2,,,2 22 yx
sec–1(secx) is periodic function with period
CHAPTER 6: Inverse Trigonometric Functions
Solved example
5. What is the value of 1 cos(cos10) ?
Sol. We know that 1 cos(cos),if0θθθπ =≤≤ Here, 10 θ = radians do not lie between 0 and π. However ,(410) π lies between 0 and π such that cos(4 p – 10) = cos10.
11 cos(cos10)cos(cos(410))410 ππ ∴=−=−
Try yourself:
5. What is the value of 1 sin(sin7) ? Answer: 7 – 2π
6.2.5 Interconvertion between Different Inverse Trigonometric Functions
■ Each inverse trigonometric function can be expressed in terms of remaining functions. ■ () 12 112 1 2 cos1if01 sincos1if10 tanif1,1 1 xx xxx x x x
) ) 12 12 2 11 2 1 sin1if0,1 sin1if1,0 1 costanif0,1 1 tanif1,0 xx xx x xx x x x x
1 2 1 1 2 sinfor0 tan1 1 cosfor0 1 x x x x x x
Solved example
6. What is the value of
Try yourself:
6. Find the value of tan{sin –1 (cos(sin –1 x ))} tan{cos–1(sin(cos–1 x))}.
Answer: 1
6.2.6 Identities of Complementary Angles
■ A pair of angles are said to be complementary angles if their sum is 90°.
Try yourself:
7. Find the minimum and maximum values of f(x)= sin–1x + cos–1x + tan–1 x.
Answer: min(()) 4 fx π = , max 3(()) 4 fx π
6.2.7 Formulae for Sum and Difference of Inverse Trigonometric Function (tan–1 x, cot–1 x)
■ Using the following formulae, sum of inverse trigonometric functions can be evaluated.
12. If –1 ≤ x ≤ 0, then what is the value of ()12 cos21 x ?
Answer: 2π-2cos–1x
CHAPTER 6: Inverse Trigonometric Functions
6.2.12 Inverse Trigonometric Functions of Multiple Angles of tan–1 x
■ Inverse trigonometric functions involving multiple angles of tan–1 x can be simplified using the following formulae.
■ 11 2 2 2tantan 1 x x x = , if 11 x −<< . ■ 11 2 2 2tantan 1 x x x π
, if 1 x > . ■ 11 2 2 2tantan 1 x x x π
1 x <− .
x −≤≤ .
■ 11 2 2 2tansin 1 x x x π
=− + , if 1 x > .
■ 11 2 2 2tansin 1 x x x π
=−− + , if 1 x <−
Solved example
Try yourself:
13. Find the value of
TEST YOURSELF
1. The solution of the inequality 11 11 22 logsinlogcos
(2) 1 ,1 2 x ∈ (3) 1 0, 2 x ∈ (4) 1 0, 2 x ∈
2. The number of real values of x satisfying the equation ()() () 3311 113 sincos 7 tancot xx xx + = + is (1) 0 (2) 1 (3) 2 (4) 3
3. For n ∈ N, if 1111 1111 tantantantan, 3454 n +++= π then n is equal to (1) 43 (2) 47 (3) 49 (4) 51
4. If 115 coscos 12 xy ab π += and 11sinsin 12 xy ab π −= , then 22 22 xy ab + is (1) 1 (2) 1 4 (3) 3 4 (4) 5 4
5. sin –1(sin 3) + sin –1(sin 4) + sin –1(sin 5) is equal to (1) –1 (2) –2 (3) 12 (4) –2 p
CHAPTER 6: Inverse Trigonometric Functions
6. If f ( x ) = 2 tan –1x + 1 2 2 sin 1 x x + , x > 1, then f(5) is equal to (1) 2 π (2) p (3) 4 tan–1(5) (4) tan165 156
7. If tan13 2 x A kx = and tan12 3 xk B k = , then A – B =
(1) 4 π (2) 2 π (3) 6 π (4) 3 π
8. If 111 35 sinsinsin 513 x +=
, then the value of x is (1) 56 65 (2) 64 65 (3) 16 65 (4) 65 66
9. 1 1 41 cos 49 2 sin 7
=
(1) 1 (2) 2 (3) 3 (4) 4
10. If x,y, and z are in AP and tan–1x , tan –1y, and tan–1z are also in AP, then (1) x = y = z (2) 2x = 3y = 6z (3) 6x = 3y = 2z (4) 6x = 4y = 3z
11. 1111 tancostancos 4242 aa bb ππ ++−
=
(1) b/a (2) a/b (3) 2 a/b (4) 2 b/a
Answer Key
(1) 3 (2) 2 (3) 2 (4) 4
(5) 2 (6) 2 (7) 3 (8) 1
(9) 2 (10) 1 (11) 4
6.3 SOLVING INVERSE TRIGONOMETRIC EQUATIONS
■ Equations involving inverse trigonometric functions can be solved by using the identities discussed. We must ensure that the roots obtained satisfy the given equation. Principal values must be taken into consideration while solving.
■ At times, we can solve problems containing one or more variables by using graphs.
Solved example
14. Find the number of positive integral solutions of the equation
2 3 tancossin. 110 xy y += +
TEST YOURSELF
1. Number of common points for the curves y = sin –1 (2 x ) + tan11 2 x
+ 2 a nd y = cos–1(2x + 5) + 1 is (where [.] denotes greatest integer function) (1) 0 (2) 1 (3) 3 (4) 4
2. The number of real solutions of () 1121sin1 2 tan xxxx π ++++= is (1) 0 (2) 1 (3) 2 (4) infinite
3. The number of solutions of the equation sin–1x = 2tan–1x is (1) 3 (2) 1 (3) 4 (4) 2
4. If sin –1x + cos –1(1 – x ) = sin –1(– x ), then x satisfies the equation (1) 2x2 –x + 2 = 0 (2) 2x2 –x = 0 (3) 2x2 + x + 1 = 0 (4) 2
5. Total number of ordered pairs ( x,y ) satisfying |y| = cosx and y = sin–1(sinx), where x ∈ [0, 3π], is equal to (1) 1 (2) 2 (3) 3 (4) 4
6. The product of all values of x satisfying the equation 2 11
As x, and y are positive integers, corresponding to x = 1, 2, we get y = 2, 7, respectively. Therefore, the given equation has 2 solutions, (,)(1,2)and (2,7) xy =
Try yourself:
14. Solve the equation Solve11theequationsin6sin63/2. xx π+=−
Answer: –1/12
Answer
CHAPTER 6: Inverse Trigonometric Functions
6.4 TELESCOPIC SERIES
■ Suppose, a given expression contains the summation ()() 1 1 n k fkfk = ∑+−
■ By substituting the values of k, the summation reduces to ()() 11fnf +− , e.g., ()111 1 1 11 1 tantantan 1 nn rr rr rr rr xx xx xx == =− +
tan–1x n – tan–1x0, ∀ n ∈ N
Solved example
15. Simplify 1 2 1 1 tan 2 n rr = =∑ . Sol. ()() ()222 122121 24141 rr rrr +−− == +− (()() ) 11 1 sumtan21tan21 n r rr = =+−−
()() 11 tan21tan1 r =+− tan1 1 n n = +
Try yourself:
15. If 1111 tantan 121(2)(3) ++ ++ 11 1 tantan, 1(1) x nn …+= ++ then what is the value of x? Answer: 2 n n +
TEST YOURSELF
1. Let a1 = 1, a2, a3, a4,….... be consecutive natural numbers. Then, 11 1223 11 tantan 11aaaa
(3) 1 tan2022 4 π (4) () 1 cot2022 4 π
2. The sum of infinite terms of the series 111124 tantantan 3933 +++
....∞ is (1) 4 π (2) 2 π (3) p (4) 2 p
3. If 111444 tantantan 71939 +++
11 4 tan......tan 67 k +∞= , then k = (1) 3 (2) 1 (3) 2 (4) 1 2
4. The sum 1 2 1 3 tan 1 nnn ∞ = +− ∑ is equal to (1) 31cot2 4 π + (2) 1 cot3 2 π + (3) p (4) 1 tan2 2 π +
5. For any positive integer, define fn: (0, ∞) → R as ()() () 1 1 1 tan 11 n n J fxxjxj = +++− ∑ for all x ∈ (0, ∞). Here, the inverse trigonometric function tan–1x assumes values in , 22 ππ . Then, which of the following
statements is true?
(1) (() ) 2 1 tan055 n j j f = ∑=
(2) (() )()() 10 12 1 10sec010 jj J ff = ∑+=
(3) For any fixed positive integer n, (() ) limtan1 n x fx n →∞ =
(4) For any fixed positive integer n, (() ) 2 limsec1 n x fx →∞ =
Answer Key
(1) 1 (2) 1 (3) 3 (4) 1
(5) 4
6.5 STANDARD RESULTS
■ In solving problems related to inverse trigonometric functions in differentiation and integration, the following substitutions are useful.
Expression Substitution Domain
1 22 ax sin xa=θ , 22
2 22 ax + tan xa=θ ,
22
■ Range of some special inverse trigonometric functions:
()() 33 33 117sincos 328 xx ππ ≤+≤
()() 22 22 115sincos 84 xx ππ ≤+≤
()() 22 22 113cossin 44 xx ππ −≤−≤
Important points to remember:
■ If 111 tantantan,f1 2 xyzixyyzzx π ++=++= then xy + yz+zx = 1
■ If 111 tantantan, xyz ++=π then x+y+z=xyz.
■ If 11tantan 2 ab xx π += then xab = .
■ If 11sinsin 2 ab xx π += then 22xab =+
■ If 11tantan 2 xy π += , then 1 xy = .
■ If 11cotcot 2 xy π += , then 1 xy = .
■ 11tantan 4 pqp qqp π += + .
■ 11 1 tantan,if1 14 x xx x +π
.
■ 11 1 tantan,if1 14 x xx x π =−>− + .
■ If 111 coscoscos3 xyz ++=π then, 1 xyz===−
■ If 1113sinsinsin 2 xyz π ++= then, 1 xyz===
■ If sin–1x + sin–1y = q then cos–1x + cos–1y = π – q.
■ If 11sincos axbxc −= , then () 11sincos abcab axbx ab π+− += +
■ Infinite series of inverse trigonometric functions: 3 157 1.31.3.5 sin...... 2.32.4.52.4.6.7 x xxxx =++++ 357 tan1 357 xxx xx=−+−+
# Exercises
JEE MAIN
Level – I
Domain and Range of Inverse Trigonometric Functions
Single Option Correct MCQs
1. The range of the function, f ( x ) = cot –1x + sec–1x + cosec–1x, is (1) 3 , 22
3 ,, 22
(4) 3 ,, 22
2. The value of
3. If 1 cot,, 6 n nN
CHAPTER 6: Inverse Trigonometric Functions
4. Range of f(x) = sin–1x + tan–1x + sec–1x is (1) 3 , 44
(3) 3 , 44 ππ
(2) 3 , 44
(4) , 43
5. If cos –1 α + cos –1 β + cos –1 γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals to (1) 0 (2) 1 (3) 6 (4) 12
Numerical Value Questions
6. If domain of the function ()() 3 11 2 cossinlog1 2 fxxx =++ is [a,b], then a + 2b =________.
Properties of Inverse Trigonometric Functions
Single Option Correct MCQs
7. Tan–12, Tan–13 are two angles of a triangle, then the third angle is:
(1) 30° (2) 45° (3) 60° (4) 75°
8. If 111 35 sinsinsinx 513 +=
then the value of x is (1) 56 65 (2) 64 65 (3) 16 65 (4) 65 66
9. If sin–1 x + 4 cos–1 x = p , then x = (1) 1 2 (2) 1 2 (3) 3 2 (4) 1
10. 11132 cotsinsintan 173
(1) 2 13 (2) 0 (3) 2 13 (4) 2 313
11. 1111 2tantan 37 += (1) p (2) 2 π
(3) 4 π (4) 3 4 π
12. If cos–1x – cos–1 2 y = α, then 4x2 – 4xy cosα + y2 is equal to (1) 2sin 2α (2) 4 (3) 4 sin2α (4) – 4 sin2α
13. Let () 1 2 2 sin 1 fxx x = + and () 2 1 2 1 cos 1 x gx x = + , then the value of (f(10) – g(100)) is equal to (1) π – 2(tan–1(10) + tan–1(100)) (2) 0
14. If the value of sin(cot –1 (cos(tan –1 x ))) is
2 xa xb + + , then a + b = (1) 1 (2) 3 (3) 4 (4) 5
15. If α and β are the roots of the equation x2 + mx – m + 1 = 0 then the principal value of tan–1α + tan–1β = (1) 4 π (2) 4
16. If 22 1 22 11 tan 11 xx xx α +−− =
then x2 =
(1) sin(2α) (2) cos(2α) (3) tan(2α) (4) cot(2α)
17. If 1131 cos,tan 53αβ
, 0 < α, 2 βπ < , then α – β =___ (1) tan19 14
(2) 19 cos 510
(3) sin19 510
(4) tan19 510
18. The value of 1 15 tancos 23
is (1) 35 2 + (2) 35 + (3) () 1 35 2 (4) 23
19. If 111477 sincostan0, 17536 α +−= 0 < α < 13, then sin–1(sinα) + cos–1(cosα) is equal to (1) p (2) 0 (3) 16 (4) 16 – 5 p
20. Given 1 0 2 x ≤≤ . Then the value of 2 11 1 tansinsin 22 xx x
+−
is (1) –1 (2) 1 (3) 1 3 (4) 3
21. If x = sin(2tan–12), 1 14 sintan, 23 y = then (1) x = 1 – y (2) x2 = 1 – y (3) x2 = 1 + y (4) y2 = 1 – x
22. The value of 111 3424 coscostansintan 10353
is (1) 0 (2) 4 π (3) 3 π (4) 6 π
Numerical Value Questions
23. sec2(tan–12) + cosec2(cot–13) is equal to____.
24. If α = sin(cot–1(tan(cos–1 2 3 ))) and β = sin (cosec–1(cot(tan–1 1 3 ))) are the roots of the quadratic equation ax2 + bx + c = 0, where a,b, and c are integers and c is prime, then the value of (a + b + c) equals _____.
Solving Inverse Trigonometric Equations
Single Option Correct MCQs
25. If π +=
11tantan 2 ab xx then x is (1)
26. If
22 sinsin2tan, 11 ab x ab then x is (1) + 1 ab ab (2) + 1 b ab (3) 1 b ab (4) + 1 ab ab
27. If π +=11 cos2cos3 3 xx then x is (1) 3 27 (2)
28. The number of positive solutions satisfying the equation
CHAPTER 6: Inverse Trigonometric Functions
++
is _____. (1) 0 (2) 1 (3) 2 (4) 3
29. If 2tan–1(cosx) = tan–1(2cosecx), then x = (1) 4 π (2) 6 π
(3) 2 π (4) no solution
30. All x satisfying (cot–1x)2 – 7(cot–1x) + 10 > 0 lie in the interval (1) (cot 2, ∞) (2) ( ∞ , cot 5) ∪ (cot 2, ∞)
31. The number of positive integral solutions of the equation tan–1x + cot–1y = tan–13 is (1) 0 (2) 1 (3) 2 (4) 3
32. If sin–1(x) + sin–1(2x) = 3 π then x = (1) 3 28 (2) 3 28 (3) 3 28 (4) 3 28
33. The number of solutions of the equation tan–1(1 + x)) + tan–1(1 – x) = 2 π is (1) 3 (2) 2 (3) 1 (4) 0
Numerical Value Questions
34. If 1111 tantan 1 xx xx +− +=+
π tan–1 (–7) then x =___
Telescopic Series
Single Option Correct MCQs
35. If a1, a2, a3 a n are in AP, with common difference
Numerical Value Questions
40. If S is the sum of the first 10 terms of the series
36.
38.
, then 12 tan(S) is equal to _______.
Level – II
Domain and Range of Inverse Trigonometric Functions
Single Option Correct MCQs
1. Let 1 51 tansin2cos 165
and 1145 cossinsec 53
, where the inverse trigonometric functions take principal values. Then, the equation whose roots are α and b is (1) 15x2 – 8x –7 = 0
(2) 5x2 – 12x +7 = 0
(3) 25x2 – 18x –7 = 0 (4) 25x2 – 32x +7 = 0
2. The range of the function f(x) = sin–1[x2 + 1 2 ] + cos–1[x2 –1 2 ] (where [.] is GIF) is (1) 2 π
(2) { p }
(3) 1 ,0 2
(4) 0, 2 π
3. The maximum value of
2 1 42 122 tan 23 x fx xx = ++ is (1) 18° (2) 36° (3) 22.5° (4) 15°
4. If [sin–1(cos–1(sin–1(tan–1x)))] = 1, where [.] denotes the greatest integer function, then x is given by the interval (1) [tan(sin(cos 1)), tan(sin(cos(sin 1)))] (2) (tan(sin(cos 1)), tan(sin(cos(sin 1)))) (3) [–1, 1]
(4) [sin(cos(tan 1)), sin(cos(sin(tan 1)))]
Numerical Value Questions
5. Let f(x) = [tan–1sin–1x + sin–1tan–1x], where [ x ] denotes the greatest integer function. Then the number of integers in the range of f is ______.
Properties of Inverse Trigonometric Functions
Single Option Correct MCQs
6. If 46 sin12...... 39 xx x −++ 812 14 cos....., 392 xx x π −+=
where
03, x ≤< then the number of values of ‘ x’ is equal to (1) 1 (2) 2 (3) 3 (4) 4
7. If p < q < r < 0, then cot11 pq pq + = Σ (1) 0 (2) p (3) 2 p (4) 2 π
CHAPTER 6: Inverse Trigonometric Functions
8. If x 1 , x 2 , x 3 , and x 4 are the roots of the equation x4 – x3 sin 2β + x2 cos 2β – xcos β – sinβ = 0, then the value of tan–1x1 + tan–1x2 + tan–1x3 + tan–1x4 is (1) 2 π (2) 2 πβ
10. x,y ∈ R,x ≠ y, and x ≠ 1. If ax + bsec(tan–1x)
= c and ay + bsec(tan–1y) = c, then 1 xy xy + =
(1) 22 2ab ab (2) 22 2ac ac +
(3) 22 2ab ab + (4) 22 2ac ac
11. The value of tan–1( 1 2 (tan2A) + tan–1(cotA) + tan–1(cot3A)) is
(1) 0,if 42 A ππ << (2) ,if0 4 A π π<<
(3) both (1) and (2) (4) None of these
12. If a,b,c are positive, then () tan1 aabc bc ++ ∑=
(1) π (2) 3 2 π
(3) 3 4 π (4) 3
13. If m and M are the least and greatest values of (cos–1x)2 + (sin–1x)2, then M m =
(1) 10 (2) 5 (3) 4 (4) 2
14. If x1, x2, and x3 are the roots of x3 – 6x2 + 11x – 6 = 0, then cot–1(x1) + cot–1(x2) + cot–1(x3) is equal to (1) 0 (2) 2 π (3) π (4) 3 2 π
Numerical Value Questions
15. If for x < –1, 1 2 2 sin 1 x x + = k π – 2tan –1 x , then 3 – k =______.
16. If tan–1x + 2 tan11 3 y y π = and sin–1y –1 2 cos 16 x x π = + , then 1 1 5sin sin x y is ____.
17. Let y = sin–1(sin6) – tan–1(tan8) + cos–1(cos6) simplifies to aπ + b, then (a – b) is equal to (a,b ∈ I)_____.
Solving Inverse Trigonometric Equations
Single Option Correct MCQs
18. The number of solutions of the equation 12122 12 sincos, 33 xxx ++−=
for x ∈ [–1, 1] where [x] denotes the greatest integer less than or equal to x, is
(1) Infinite (2) 2 (3) 4 (4) 0
19. The number of positive integral solutions of the equation
111 2 3 tancossin 110 xy y += + is
(1) one (2) two (3) zero (4) none of these
20. The value of x for which 2 1 2 24 sinsin3 1 x x π +
is
(1) (0, 1)
(2) (–1, 0)
(3) (–1, 1)
(4) (0, 2)
21. The sum of roots of the equation cos–1(cosx) = [x] (where [.] is GIF) is (1) 2π + 3 (2) π + 3 (3) π – 3 (4) 2π – 3
22. The complete solution set of [tan –1 x ] 2 – 8[tan –1 x ] + 16 ≤ 0, where [ ] denotes the greatest integer function, is (1) (– ∞ , tan 4]
(2) [tan 4, tan 3]
(3) (tan 5, ∞ )
(4) No Solution
Numerical Value Questions
23. Number of integral values of k, such that the equation cos–1x + cot–1x = k possesses solution, is _____.
24. The number of solutions of the equation |cosx| = sin–1|sinx| in [0, 2π] is ____.
25. The number of solutions of the equation, tan–1(4{x}) + cot–1(x + [x]) = 2 π is ___.
(Note: [ ] an d {} denote greatest integer function and fractional part function, respectively.]
26. Number of integral values satisfying the inequality 2 1 2 24 sinsin3 1 x x π
+
+
is
Telescopic Series
Single Option Correct MCQs 27. 121212333 cot1cot2cot3
is equal to (1) () tan1 4 n π (2) () 1 tan1 4 n π +− (3) () tan1 n (4) () 1 tan1 n +
30. Value of 42 1 1 limcot1 2 n n r rr r →∞ =
CHAPTER 6: Inverse Trigonometric Functions
Numerical Value Questions
31. If the value of the series () 11 42 1 8 taniscot 25 n nk nn ∞ = −+ ∑ then |k| is equal to ___.
32. If 1 2 1 18 tan, n k n π λ ∞ = =∑ where k and λ are natural numbers that are coprime, then |k – λ| is ___.
Level – III
Single Option Correct MCQs
1. 33 2121 11 cosectansectan 2222 +
is equal to
(1) (α – β)(α2 + β2) (2) (α + β)(α2 – β2)
(3) (α + β)(α2 + β2) (4) none of these
2. The number of values of a ∈ R, for which the equation 5tan–1(x2 + x + a) + 3cot–1(x2 + x + a) = 2π has a unique solution, is (1) 0 (2) 1 (3) 2 (4) infinity
3. The number of solutions of (()() ) 11 75 loglogtantan5tantan0 xx++= in [0, 3π] is (1) 0 (2) 4 (3) 6 (4) 3
4. The number of real solutions of the equation 1212 tan32cos43 xxxx−++−−=π
(1) one (2) two (3) zero (4) infinite
5. If y ∈ (–3, 0), 2 11 2 692 tancot 963 yy yy π += then 33 y =_____ (1) 3 (2) 5 (3) 6 (4) 9
7. 1 42 1 2 tan 2 n m m mm = ++ ∑ is equal to
(1)
2 1 2 tan 2 nn nn + ++
(2) 2 1 2 tan 2 nn nn −+
(3) 2 1 2 2 tan nn nn ++ + (4) 2 1 2 2 tan nn nn ++
Numerical Value Questions
8. If the equation sin–1(x2 + x + 1) + cos–1(λx + 1) = 2 π has exactly two solutions for λ ∈ [a,b), then the value of a + b is_________
10. If () () 1 223 1 21 tantan 112 n r r rrrrr =
++−+−
= 961, then the value of n is equal to ____.
11. If f(x) = (cos–1x)2 – (sin–1x)2, the sum of all the possible integral values of () 2 4 fx π is _____.
12. The value of 1 π {216 sin –1 (sin 7 6 π ) + 27 cos –1 (cos 2 3 π ) + 28 ta n –1 (tan 5 4 π ) +
200 cot–1(cot 4 π )} must be ___.
13. The least positive integral value of k, for which (k – 2)x2 + 8x + k + 4 > sin–1(sin12) + cos–1(cos12) for all x ∈ R, is _____.
14. If x2 + y2 + z2 = r2, then 111 tantantan. xyyzxzK zrxryr
Then, [k] = ____, where [. ] is GIF.
15. Find the maximum value of x for which 2 11 2 1 2tancos 1 x x x + + is independent of x
THEORY-BASED QUESTIONS
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,
9. Considering only the principal values of the inverse trigonometric functions, the value of 111 22 321222 cossintan 2422 π πππ ++ ++ is k. Then [k] = _________. (where [.]denotes GIF).
(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,
1. S-I : If 222,0,0 abccab+=≠≠ , then the non zero solution of the equation 111 sinsinsinis1 axbx x cc +=± .
S-II : () 111 sinsinsinxyxy +=+ .
2. S-I : () 11 tansin1,1 xyy=⇒∈− .
S-II : Minimum value of ()()22211 sincosis 8 xx π +
3. S-I : If sin1,1 x ∈ , then 11sincosxx > .
S-II : () 1 coscos1,1 xxx=∀∈− .
4. S-I : () 1 tansectan. 42 π +=+ x xx
S-II : 1 tan1tan1 > .
5. S-I : If 11sin2sinxa = , then the range of A is 11 , 22
S- II : If 1111 secsecsecsec then. +=− = xx ba ab xab
6. S-I : 11 sincos1 5 x = has no solution.
S-II : The equation 1 11 cossin 52 x = has no solution.
CHAPTER 6: Inverse Trigonometric Functions
8. S-I : 1111 tantan 1 xx xx +− +
= π + tan–1(–7) ⇒ x = 2.
S-II : 1111 tantan 1 xx xx +− +
= tan–1(–7) ⇒ x = 2.
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.
9. (A) : The value of 111tantan 2 π += x x for all {} 0 xR∈−
(R): 1 1 1 1cotfor0 tan cotfor0 >
10. (A) : The solution set for () 12 coscos434xx>− is φ
(R) : The value of cos–1(cos x) = 2π – x for x ∈ (π, 2π).
11. (A) : The value of 2 11331 coscoswhen1 2232 xx
11331 coscoswhen1 2232
7. S-I : tan–1(cotx) + cot–1(tanx) = π – 2x.
S-II : If 2 11 22 21 3sin3cos 11 ++ xx xx , then 1 3 x =
(R) : 1 cos x is increasing function for 01 x ≤≤
12. (A) : 111 tan1tan2tan3++=π
(R) : tan–1 x + tan–1 y = 1 11 1 tan,if<1 1 tantan tan,if1 1 π
xy xy xy xy xy xy xy
18. (A) : sin–1(2cos2x – 1) + cos–1(1 – 2sin2 x)
()() 1212 sin2cos1cos12sin 2 xx π −+−=
(R) : [] 11 sincos1,1 2 xxx π +=∀∈−
13. (A) : ()() 11 sin2sinsincos2 2 xx π += ()() 11 sin2sinsincos2 2 xx π +=
13. If S n = cot–1(3) + cot–1(7) + cot–1(13) + cot–1 (21) +……...n terms, then (1) 1 10 5 tan 6 S = (2) 4 S π ∞ = (3) 1 6 4 sin 5 S = (4) S20 = cot–11.1
14. If a ≤ sin–1x + cos–1x + tan–1x ≤ b, then (1) 4 a π = (2) a = 0 (3) b = π (4) 3 4 b π =
15. If (sin–1x)2 + (sin–1y)2 + (sin–1z)2 = 32 4 π , then the value of (x – y + z) can be (1) 1 (2) –1 (3) 3 (4) –3
Numerical/Integer Value Questions
16. The total number of ordered pairs (x,y) which satisfies y = |sin x |, y = cos –1(cos x ), where –2π ≤ x ≤ 2π, is ___.
17. The number of real solutions of the equation 11 11 sin 22 i i ii x xx π ∞∞ −+ ==
() 1 11 cos 2 i i ii x x ∞∞ ==
lying in the interval (0, 2) is (Here, the inverse trigonometric function sin –1x and cos –1 x assume values in , 22 ππ
and [0, π], respectively).
18. Number of solution(s) of the equation () 7113 tancot 2 xxx π π ++= for x ∈(1,2) is ___.
19. Let () tan1, 22 x ππ ∈−
, for x ∈ . Then the number of real solutions of the equation ()() 1 1cos22tantan xx+= in the set 33 ,,, 222222 ππππππ −−∪−∪ is equal to ____.
20. If 1 cot, 6 n nN π π >∈ , then the maximum value of n is ____.
21. If sin–1x + sin–1y + sin–1z = π, then the value of 444222 222222 4 xyzxyz xyyzzx +++ ++ is ___.
22. sec2(tan–12) + cosec2(cot–12) = ____.
23. 1 cot2cot3......... 4 π −= ____.
24. If the equation sin–1(x2 + x + 1) + cos–1 (λx + 1) = 2 π has exactly two solution for λ ∈ [a,b], then the value of a + b is _____.
25. If 115sincosec 542 x += π , then the value of x = ___.
26. If f ( x ) = x 3 – 3 x + sin –1( a 2 – 3 a + 2), then the smallest positive integer a for which f(x) = 0 has three distinct real solutions is ___.
27. Let α, β, and γ be the roots of the equation x3 + 6x + 3 = 0 and A = cos–1(sin((α + β)–1 + (β + γ)–1 + (γ + α)–1 )), 1 costansin, 2 B αβγ
C = sec–1(cosec((1 – α)(1 – β)(1 – γ))). Then the value of (5A + 2B – C) is equal to ____.
28. If 111 3648 sin,cos,tan 85515αβγ
and cossinsincossinsincossinsin , coscoscos A αβγβαγγβα αβγ ++ =
B = tanα∙ tanβ + tanβ ∙ tanγ + tanγ tanα, then the value of A + B is ___.
29. If the domain of the function ()() 1 3cos4 fxx=π is [a,b], then the value of (4a + 64b) is ____.
30. If maximum value of 2tan –1 x + sin –1 x + 3sec–1x is pπ, then p = ____.
Passage-based Questions
(Q.31 – 32)
Consider the equation 1111 tancostancos1 4242 xx ππ ++−=
31. Number of solutions of the above equation in (0, 1) is _______.
CHAPTER 6: Inverse Trigonometric Functions
32. Number of solutions of the above equation in (–1, 0) is _____.
(Q.33 – 34)
Let 1111 tantan 23 α=+ , and b = tan–1 1 + tan–1 2 + tan–1 3
33. The value of 8 α π is ___.
34. The value of 4[b] is ___. (where [.] represents GIF).
(Q.35 – 36)
+−
Let ()() 42 1 23 361 sin 1 fxxx x
+
and ()()()()() 12 12 3sec1;1 3cosec1;1 fxxx gx fxxx
.
35. The value of ()() 33 1 tan22cot15sin18 2 ggg
is (1) 2π (2) 4π (3) 7 2 π (4) 9 2 π
36. If ()() 1 3sin4cos, 5 hxxx =+ then domain and range of g(h(x)) respectively, are (1) [–1, 1]; [π, 3π] (2) R; {π} (3) 3 ; 2 R π
(4) 3 1,1;, 22 ππ
(Q.37 – 38)
Let a + bcos–1x = cos–1(4x3 – 3x).
37. If 1 1, 2 x ∈−
, then the value of a + bπ is of the form Kπ. Then, K = ___.
38. If 1 ,1 2 x ∈ , then the value of 0 cos y Ltby → = _____.
(Q.39 – 40)
Let S n = cot –1 (3) + cot –1 (7) + cot –1 (13) + cot–1(21) + ...n terms
39. If S k π ∞ = , then k =______.
40. If 1 10 5 tan S p = , then p =_____.
(Q.41 – 42)
Let ax + b (sec(tan –1 x ) = c and ay + b (sec (tan–1 y) = c. Then
41. The value of xy is (1) 22 2ab ab (2) 22 22 cb ab (3) 22 22 cb ab + (4) none of these
42. The value of x + y is (1) 22 2ac ab (2) 22 22 cb ab (3) 22 22 cb ab + (4) none of these
(Q.43 – 44)
For x,y,z,t ∈ R , sin –1x + cos –1y + sec –1z ≥ t2 – 2 tπ + 3π.
43. The value of x + y + z is equal to (1) 1 (2) 0 (3) 2 (4) –1
44. The principal value of cos –1(cos5t2) is (1) 3 2 π
(Q.45 – 47)
Consider the system of equations cos –1 x + (sin–1y)2= 2 4 pπ and (cos–1x)(sin–1y)2= 4 , 16 π p ∈ Z.
45. The value of p for which the system has a solution, is (1) 1 (2) 2 (3) 0 (4) –1
46. The value of x which satisfies the system of equation is (1) 2 cos
47. Which of the following is not the value of y that satisfies the system of equations? (1) 1 (2) –1 (3) 1 2 (4) 1, –1
(Q.48 – 50) If 1111 tantan 23 α=+
, then
48. cos ( a+b+γ)= (1) 5 cos
49. tantan3tan 24 αβγ
(1) 4 (2) 3 (3) 2 (4) 1
50. sin(cot–1(tan(cos–1(sinγ)))) = (1) sinγ
(2) sin 2
(3) 1 sin 2 γ
(4) cosγ
(Q.51 - 53)
51. The sum of infinite terms of the series
4
52. The value of
53. The sum of infinite terms of the series
CHAPTER 6: Inverse Trigonometric Functions
(1) 4 π (2) 2 π (3) cot–12 (4) –cot–12
(Q.54 – 56)
While defining inverse trigonometric functions, a new system is followed where domains and ranges have been redefined as follows:
sin–1x [–1, 1] 3 , 22
tan1xR 3 , 22
cos–1(x) [–1, 1] [π, 2π] cot–1xR [π, 2π]
54. sin–1(–x) is equal to (1) – sin–1x (2) π + sin–1x (3) 2π – sin–1x (4) 12 3cos1,0 xx π−−>
55. If f(x) = 3sin–1x – 2cos–1x, then f(x) is (1) even function (2) odd function (3) neither even nor odd function (4) even as well as odd function
56. The minimum of (sin–1x)3 – (cos–1x)3 is equal to (1) 633 8 π (2) 633 8 π (3) 1253 32 π (4) 1253 32 π
(Q.57 – 58)
If tan(tan–1x) = x ∀ x ∈ , cot(cot–1x) = x ∀ x ∈
57. 2cot(cot–1(3) + cot–1(7) + cot–1(13) + cot –1 (21)) has the value equal to ________.
58. If
, m n = where m,n ∈ N, then the least value of ( m + n) is ____.
Matrix Matching Questions
59. Match the following and choose the correct option
List–I List–II (A) If sin–1x+Sin–12x = 3 π , then x= (p) 3 2 (B) If tan –1 2 x +tan –1 3 x = 4 π , then x= (q) 13 (C) If sin–1x–cos–1x= 6 π , then x= (r) 1 6 (D) If 11512 sinsin 2 xx π += , then x = (s) 3 28 (A) (B) (C) (D) (1) s r q p
q r s p
s r p q (4) q r p s
60. Match the following and choose the correct option.
List I List II (A) sinsin11 32 π
(p) 35 2 (B) 13 coscos 26 π
−+
(C) 1 15 tancos 23
(D) tan2sin14 5
(A) (B) (C) (D)
(1) r s p q
(2) s q p r
(3) p r q s
(4) r s q p
(q) 24 7
(r) 1
(s) –1
61. Match the following and choose the correct option.
List I List II
(A) x ∈ [π,2π] ⇒ |tan–1(tanx)|can be (p) |x – 2π|
(B) x ∈ [π, 2π] ⇒ |cot–1(cotx)| can be (q) |x – π|
(C) x ∈ [–π,π] ⇒ |sin–1(sinx)|can be (r) |x |
(D) x ∈ [–π,π] ⇒ |cos–1(cosx)|can be (s) |x + π|
(A) (B) (C) (D)
(1) r s p q
(2) p,q p q,r,s r
(3) q p,r s r
(4) p,q p,s r,s r
62. Match the following and choose the correct option.
List I
List II
(A) 1141 sin2tan 53 += (p) 6 π
(B) 111 12463 sincostan 13516 ++ (q) 2 π
(C) If tan13 2 x A x λ = and tan12 3 x B λ λ = , then the value of A–B is, (r) 4 π
(D) 1111 tan2tan 73 + (s) π
(A) (B) (C) (D)
(1) s q p r
(2) q s p r
(3) r s q p
(4) p s r q
FLASHBACK (P revious JEE Q uestions )
JEE Main
1. Considering only the principal values of inverse trigonometric functions, the number of positive real values of x satisfying ()() 11 tantan2 4 xx π += is
(27th Jan 2024 Shift 2)
(1) More than 2 (2) 1 (3) 2 (4) 0
2. For α, β, γ ≠ 0, if sin–1α + sin–1β + sin–1γ = π and (α + β + γ)(α – γ + β) = 3αβ, then γ is equal to (31st Jan 2024 Shift 1)
(1) 3 2 (2) 1 2
CHAPTER 6: Inverse Trigonometric Functions
(3) 31 22 (4) 3
3. If a = sin–1(sin(5)) and b = cos–1(cos(5)), then a2 + b2 is equal to (31st Jan 2024 Shift 2)
(1) 4π2 + 25
(2) 8π2 – 40π + 50
(3) 4π2 – 20π + 50
(4) 25
4. Let x * y = x2 + y3 and (x*1)* 1 = x* (1* 1). Then the value of 42 1 42 2 2sin 2 xx xx +− ++ is (24 th Jun 2022 Shift 2) (1) 4 π (2) 3 π (3) 2 π (4) 6 π
5. The set of all values of k for which (tan–1x)3 + (cot–1x)3 = kπ3, x ∈ R, is the interval (24th Jun 2022 Shift 1)
(1) 17 , 328
(3) 113 , 4816
(2) 113 , 2416
(4) 19 , 328
6. The value of 1 i ta 15 cos1 4 n sn 4 π π
is equal to (25th Jun 2022 Shift 2) (1) 4 π (2) 8 π (3) 5 12 π (4) 4 9 π
7. If the inverse trigonometric functions take principal values, then
8. Let f(x) = 2cos–1x + 4cot–1x – 3x2 – 2x + 10, x ∈ [–1, 1]. If [ a,b ] is the range of the function, then 4a – b is equal to (26 th June 2022 Shift 1) (1) 11 (2) 11
9. The value of
The value
is e qua l to _________. (29th June 2022 Shift 1)
13. Let x = sin(2tan–1α) and 1 14 sintan. 23 y =
If S = { α ∈ R : y 2 = 1 – x }, the n 163 S
α ∈ ∑ is equal to _____. (25th July 2022 Shift 2)
14. If 1 0 2 x << and 11sincosxx αβ = , then the value of 2 sin πα αβ + is (26 th July 2022 Shift 2)
(1) ()() 224112xx
(2) ()() 224112 xxx
(3) ()() 222114 xxx
(4) ()() 224114xx
15. 111 151 tan2tansec2tan 528 ++ is equal to (26th July 2022 Shift 1) (1) 1 (2) 2 (3) 1 4 (4) 5 4
16. For k ∈ , let the solutions of the equation cos(sin–1(xcot(tan–1(cos(sin–1x))))) = k, 0 < | x | < 1 2 be α and β, where the inverse
138 trigonometric functions take only principal values. If the solutions of the equation x2 – bx – 5 = 0 are 222 11and,then b k α αββ + is ______. (27th July 2022 Shift 1)
17. Considering only the principal values of the inverse trigonometric functions, the domain of the function () 2 1 2 42 cos 3fxxx x −+ = + is (28th July 2022 Shift 1)
(1) 1 , 4
(3) 1 , 3 −∞
(2) 1 , 4
(4) 1 , 3 −∞
18. Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equaion cos–1(x) – 2sin–1(x) = cos–1(2x) is equal to (28th July 2022 Shift 1)
21. The range of () 2 1 2 4sin 1 fxx x = + , is (13th Apr 2023 Shift 2)
(1) [0, 2π] (2) [0, π] (3) [0, π) (4) [0, 2π)
22. For x ∈(–1, 1], the number of solutions of the equation sin–1x = 2 tan–1x is equal to______. (13 th Apr 2023 Shift 2)
23. 1113843 tansec 33633
is equal to (24th Jan 2023 Shift 1) (1) 2 π (2) 4 π (3) 6 π (4) 3 π
24. If the sum of all solutions of 2 11 2 21 tancot, 123 xx xx
–1 < x < 1, x ≠ 0, is 4 , 3 α then a is _____. (25th Jan 2023 Shift 1)
25. Let a 1 = 1, a 2, a 3, a 4, ........ be consecutive natural numbers. Then 11 1223 11 tantan 11aaaa
+ ++ 1 20212022 1 ..tan 1 aa +…+ + is (30th Jan 2023 Shift 2) (1) () 1 cot2022 4 π
(2) () 1 tan2022 4 π
(3) () 1 tan2022 4 π + (4) () 1 cot2022 4 π
26. Let y = f(x) represent a parabola with focus 1 ,0 2
and directrix
Then,
(1) is an infinite set
(31st Jan 2023 Shift 2)
(2) is an empty set (3) contains exactly two elements (4) contains exactly one element
27. Let (a,b) ⊂ (0, 2π) be the largest interval for which sin–1(sin θ) – cos–1(sin θ) > 0, θ ∈ (0, 2π), holds if αx2 + βx + sin–1(x2 – 6x + 10) + cos–1(x2 – 6x + 10) = 0 and α – β = b – a. Then, α is (31st Jan 2023 Shift 2) (1) 8 π (2) 48 π (3) 16 π (4) 12 π
28. Let S be the set of all solutions of the equation
(2) 2 3 π
(3) 2sin13 4 π
(4) sin13 4 π
29. Let 2 11 2 :01 and 11 2tancos 11 xRx Sxx xx ∈<<
If n(S) denotes the number of elements in S, then (1st Feb 2023 Shift 2)
(1) n(S) = 0
(2) n(S) = 1 and the element in S is less than 1 2
(3) n(S) = 1 and the element in S is more than 1 2
(4) n(S) = 2 and only one element in S is less than 1 2
JEE Advanced
30. For any y ∈ , let cot –1 ( y ) ∈ (0, π) and () tan1, 22 y ππ ∈− . Then, the sum of all the solutions of the equatio n 2 11 2 692 tancot 963 yy yy
.
Then, ()12 2sin1 xS x ∈ ∑ is equal to (1st Feb 2023 Shift 1)
2. If x1, x2, and x3 are positive roots of x3 – 6x2 + 3px – 2p = 0(p ∈ R), then the value of 1 12 11 sin xx ++
11 2331 1111 costan xxxx
+−+
is (1) 8 π (2) 6 π (3) 4 π (4) π
3. If the sum of all the solutions of 2 11 2 21 tancot, 123 xx xx π +=
–1 < x < 1, x ≠ 0 is 4 3 α , then α = (1) 1 (2) 2 (3) 3 (4) 4
4. If cos –1x – cos –1 2 y = α , where –1 ≤ x ≤ 1, –2 ≤ y ≤ 2, x ≤ 2 y for all x,y, then 4x2 – 4xy cosα + y2 is equal to (1) 2sin2α (2) 2cos2α+ 2x2y2 (3) 4sin2α (4) 4sin2α– 2x2y2
CHAPTER 6: Inverse Trigonometric Functions
5. The value of 111 444 tantantan 71939 ++ 14 tan..... 67 ++∞=
(1) 11111 tan1tantan 23 ++
(2) tan–11 + cot–13
(3) 11111 cot1cotcot 23 ++
(4) cot–11 + tan–13
6. 1 42 1 2 tan 2 n n r r Lt rr →∞ = = ++
(1) 4 π (2) 2 π (3) 4 π (4) 2 π
7. Let 1 2121 1 6 tan. 23 kr krr r S ++ = = + ∑ Then lim kk S →∞ is equal to : (1) 2 π (2) tan13 2 (3) cot13 2
(4) tan–1(3)
8. If the inverse trigonometric function takes the principal value only, then the number of real values of x, which satisfy 111 34 sinsinsin 55 xx x +=
, is equal to (1) 2 (2) 3 (3) 0 (4) 1
9. Let f ( x ) = sin x + cos x + tan x + sin –1 x + cos–1 x + tan–1x . If M and m are maximum and minimum values of f ( x ), then their arithmetic mean is equal to (1) cos1 2 π + (2) sin1 2 π + (3) tan1cos1 2 π ++ (4) tan1sin1 4 π ++
10. If 357 tan1 3!5!7!
, then the maximum value of α equals to
(1) 1 2 (2) 1 (3) (4) 1 2
11. The set of values of k, for which x 2 – kx + sin–1(sin4) > 0, for all real x, is (1) φ or null set (2) {1} (3) {1, 2} (4) {1, 2, 3}
12. The value of sin –1 (cos2) – cos –1 (sin2) + tan –1(cot4) – cot–1(tan4) + sec–1(cosec6) –cosec–1 (sec6) is (1) 0 (2) 3π (3) 8 – 3π (4) 5π – 16
13. If α is the negative real root of the equation x3 + bx2 + cx + 1 = 0 (b < c), then the value of tan–1α + tan–1 1 α = (1) 2 π (2) 2 π (3) 0 (4) π
14. IF M and m respectively, are the maximum and minimum values of the function f(x) = tan–1(sinx + cosx) in 0, 2 π , then the value of M + m =
24. If ()()222115 tancot, 8 xx π += then the absolute value of x is _______.
25. If f(x) = x11 + x9 – x7 + x3 + 1, suppose that f (sin –1 (sin1)) = a and f (tan –1 (tan(–1))) = k – a, then the value of k = ___.
CHAPTER TEST – JEE ADVANCED
2023 P1 Model Section – A [Multiple Option Correct MCQs]
1. If the ordered pairs of natural numbers (a1, b1) and (a2, b2) satisfy the equation arc cot8 = arc tan a – arc tan b (a1 < a2), then
(1) Domain of f(x) + g(x) is {1}.
(2) Domain of g(x) + h(x)is ) 2, ∞ .
(3) Domain of h(x) + f(x) is 1 ,1 2
.
(4) Domain of f(x) + g(x) + h(x) is φ.
3. If α,β, and γ are the roots of tan–1(x – 1) + tan–1x + tan–1(x + 1) = tan–13x, then
(1) α + β + γ = 0
(2) αβ + βγ + γα = 1 4
(3) αβγ = 1
(4) |α – β|max = 1
Section – B
[Single Option Correct MCQs]
4. If cot–1(y) ∈ (0, π) and 2tan–1y ∈ (–π, π) ∀ y ∈ R, then the integer nearest to the sum of solutions of the equation 2 11 2 6 93 tancot 694 yy yy π += for y > 0 is equal to
(1) 15 (2) 6 (3) 9 (4) 14
5. Let f ( x ) = sin –1 ( x ) + cos –1 ( x 2 ) + sin –1 ( x 3 ) + .....+ sin –1( x 2 n –1) + cos –1( x 2 n ) and g ( x ) = cos –1 ( x ) + sin –1 ( x 2 ) + cos –1 ( x 3 ) + ......+
cos–1(x2n–1) + sin–1(x2n). If minimum of f(x) + maximum of g(x) = 8π then n = (1) 7 (2) 8 (3) 4 (4) 16
6. If the minimum value of the function () 11sincos88 fxxx =+ is m, then the value of log2m is equal to (1) 1 4 π + (2) 3 1 4
1
7. If 111 sincostan ; xxy abc == 0 < x < 1, then the value of cos c ab π
is (1) 12 y yy (2) 1 – y2 (3) 12 2 y y (4) 2 2
Section – C [Integer Type Questions]
8. If f(x) = 2023 sin(sin–1x) + 2022 tan(tan–1x) – 2023 sin(sin–1x) – 2022 tan(tan–1x) + 2x + (2024)2 and g(x) = (2023)2 x2 – 2(2022)x + 1, then the number of value(s) of x, x ∈ R, that will satisfy the equation f(x) = g(x) is/ are _____.
9. If the sum of the series given by ()()() 111 cot22cot42cot72+++ () 1 cot112...... + infinite terms is equal to tan–1(k) then the value of [k4 + k2 + k] equals ([.] GIF) ____.
10. If the value of 888 1 111 tan pqr q pk r π ===
then greatest digit in the number k, equals
11. If f:[0, 4π] → [0, π] is defined by f(x) = cos–1 (Cos x ) then the number of points x ∈ [0, 4π] satisfying th e e quation ()10 10 fxx = is ___.
12. If y = sin(cot–1x) and x = 99, then 9802y2 = _____.
13. If 10 11 2 1 3 tancot 931 r m rrn =
(where m,n are coprime numbers), then the valu e of () 2 8 mn + is ____.
Section – D [Matrix Matching Questions]
14. Let (x,y) be such that sin–1(ax) + cos–1(y) + cos –1( bxy ) = 2 π . Match the statements in List I with List II.
List I List II
A. If a = 1 and b = 0, then (x, y) I) lies on the circle x2+y2 =1
B. If a = 1 and b = 1, then (x, y) II ) lies on (x2 – 1) (y2 – 1) = 0
C. If a = 1 and b = 2, then (x, y) III) lies on y = x
D. If a = 2 and b = 2, then (x, y) IV) lies on (4x2 – 1) (y2 – 1) = 0
Choose the correct answer from the options given below.
(A) (B) (C) (D)
(1) IV III I II
(2) II IV III I
(3) I II I IV
(4) III IV II I
15. Match List I with List II.
List I List II
CHAPTER 6: Inverse Trigonometric Functions
+
II ) 4
+
A. ()() ()() 1 1122 211 4 1costansintan cotsintansin xxx xxx x
+
takes the value(s) I) 15 23
B. If cot(sin–1 12 x ) = sin(tan–1 (6) x ), x ≠ 0, then possible values of x is/are II ) 1 2
C. If cosα + cosβ + cosγ = 0 = sinα+sinβ+sinγ, then the possible value of cos 2 αβ
is III) 2
D. If cos( 4 π – θ) cos2θ + sinθ sin2θ s ecθ = cosθ sin2θ secθ + cos( 4 π +θ) cos2θ), (θ≠ 2 nπ ,n ∈ I), then the value of secθ is IV) 1
Choose the correct answer from the options given below.
(A) (B) (C) (D)
(1) IV I III II
(2) I IV II III
(3) IV I II III
(4) I IV III II
16. Match the following List I with List II. List I List II
A. 2cot(cot–1(3)+cot–1(7)+cot –1 (13)+cot–1(21)) has the value equal to I) 3
B. If 11 1 1 11 tantan 37 1 tantan..... 13 1 tan 381
m n = , where m,n ∈ N, then sum of the digits in the least value of (m+n) is
C. Number of integral ordered pairs (x,y) satisfying the equation 11 tantanarcarc xy + 1 tan 10 arc = is
III) 5
D. The smallest positive integral value of n for which (n–2)x2 + 8x + n + 4 > sin–1(sin12)+cos–1(cos12), ∀ x ∈ R, is IV) 8
V) 10
Choose the correct answer from the options given below.
(A) (B) (C) (D)
(1) IV II I III
(2) I II II III
(3) V I IV II
(4) I IV II IV
17. If 1 ! n n r Sr = =∑ for n>6, where [.] denotes GIF and 6 1 !873 r r =
=
, then match List I with List II.
Choose the correct answer from the options given below.
