Trigonometry Formulas

Trigonometry Formula

Formula of Trigonometry:

Trigonometry is a well acknowledged name in the geometric domain of mathematics, which is in relevance in this domain since ages and is also practically applied across the number of occasions. ● ● ● ● ● ● ● ●

Integration Formula Trig Identities Formula of Trigonometry Trigonometric Ratios Trigonometric functions with Formulas What are Trigonometric derivatives Heights and distance Trigonometry formula Involving Sum Difference Product Identities ● Pythagorean Theorem ● Differentiation Formula ● Basic Trig Identities

In simple language trigonometry can be defined as that branch of algebra, which is concerned with the triangle. In this branch we basically study the relationship between angles and side length of a given triangle. With this detailed study of

triangle, several types of equations are formed, which are consequently solved to simplify the relationship between the side and angle lengths of such triangle.

Trigonometry is considered as one of the oldest components of Algebra, which has been existing around since 3rd century. There are the practical usages of trigonometry in several contexts such as in the domain of astronomy,surveying, optics or in periodic functions.

Formulas of Trigonometry

Well, whether it is algebra or geometry both of these mathematics branches are based on scientific calculations of equations and we have to learn the different formulas in order to have its easy calculation.

As we know that in Trigonometry we basically measure the different sides of a triangle, by which several equations are formed. Further the formulas of Trigonometry are drafted in accordance to the various ratios used in the domain, such as sine, tangent, cosine etc. So, basically there are the numbers of the formulas which are generally used in Trigonometry to measure the sides of the triangle.

List of All Trigonometric Formulas

Trigonometry Table Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

Angles (In Degrees)

0°

30°

45°

60°

90°

180°

270°

360°

Angles (In Radians)

0°

π/6

π/4

π/3

π/2

π

3π/2

2π

sin

0

1/2

1/√2

√3/2

1

0

-1

0

cos

1

√3/2

1/√2

1/2

0

-1

0

1

tan

0

1/√3

1

√3

∞

0

∞

0

cot

∞

√3

1

1/√3

0

∞

0

∞

csc

∞

2

√2

2/√3

1

∞

-1

∞

sec

1

2/√3

√2

2

∞

-1

∞

1

Periodicity Identities (in Radians) These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities. ● sin (π/2 – A) = cos A & cos (π/2 – A) = sin A ● sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A ● sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A

● ● ● ● ●

sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A sin (π – A) = sin A & cos (π – A) = – cos A sin (π + A) = – sin A & cos (π + A) = – cos A sin (2π – A) = – sin A & cos (2π – A) = cos A sin (2π + A) = sin A & cos (2π + A) = cos A

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.

Co-function Identities (in Degrees) The co-function or periodic identities can also be represented in degrees as: ● ● ● ● ● ●

sin(90°−x) = cos x cos(90°−x) = sin x tan(90°−x) = cot x cot(90°−x) = tan x sec(90°−x) = csc x csc(90°−x) = sec x

Sum & Difference Identities ● ● ● ● ● ●

sin(x+y) = sin(x)cos(y)+cos(x)sin(y) cos(x+y) = cos(x)cos(y)–sin(x)sin(y) tan(x+y) = (tan x + tan y)/ (1−tan x •tan y) sin(x–y) = sin(x)cos(y)–cos(x)sin(y) cos(x–y) = cos(x)cos(y) + sin(x)sin(y) tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

Double Angle Identities ● ● ● ● ● ●

sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)] cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)] cos(2x) = 2cos2(x)−1 = 1–2sin2(x) tan(2x) = [2tan(x)]/ [1−tan2(x)] sec (2x) = sec2 x/(2-sec2 x) csc (2x) = (sec x. csc x)/2

Triple Angle Identities

● Sin 3x = 3sin x – 4sin3x ● Cos 3x = 4cos3x-3cos x ● Tan 3x = [3tanx-tan3x]/[1-3tan2x]

Half Angle Identities

●

●

● ●

● sin ● x ● 2 ● =± 1−cosx ● 2 ● − ● − ● − ● − ● − ● − ● √ ● ● cos ● x ● 2 ● =± 1+cosx ● 2 ● − ● − ● − ● − ● − ● − ● √ ● tan( ● x ● 2 ● )= 1−cos(x) 1+cos(x) ● − ● −

● ● ● ● ●

− − − − √

Also, 1−cos(x) So, tan( x 2 )= 1−cos(x) sin(x)

Product identities ● sinx⋅cosy= ● sin(x+y)+sin(x−y) ● 2 ● cosx⋅cosy= ● cos(x+y)+cos(x−y) ● 2 ● sinx⋅siny= ● cos(x−y)−cos(x+y) ● 2

Sum to Product Identities ● sinx+siny=2sin ● x+y ● 2 ● cos ● x−y ● 2 ● sinx−siny=2cos

● x+y ● 2 ● sin ● x−y ● 2 ● cosx+cosy=2cos ● x+y ● 2 ● cos ● x−y ● 2 ● cosx−cosy=−2sin ● x+y ● 2 ● sin ● x−y ● 2

Inverse Trigonometry Formulas ● ● ● ● ● ●

sin-1 (–x) = – sin-1 x cos-1 (–x) = π – cos-1 x tan-1 (–x) = – tan-1 x cosec-1 (–x) = – cosec-1 x sec-1 (–x) = π – sec-1 x cot-1 (–x) = π – cot-1 x