Inverse Trigonometric Formulas

The inverse trigonometric functions are also known as the anti trigonometric functions or sometimes called as arcus functions or cyclometric functions. The inverse trigonometric functions of sine, cosine, tangent, cosecant, secant, and cotangent are used to find the angle of a triangle from any of the trigonometric functions. It is widely used in many fields like geometry, engineering, physics etc. The inverse trigonometric functions are written as sin-1x, cos-1x, cot-1 x, tan-1 x, cosec-1 x, sec-1 x. But in most of the time, the convention symbol to represent the inverse trigonometric function using arc-prefix like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). To determine the sides of a triangle when the remaining side lengths are known.

What is Inverse Trigonometric Function?

Consider, the function y = f(x), and x = g(y) then the inverse function is written as g = f-1,

This means that if y=f(x), then x = f-1(y).

Such that f(g(y))=y and g(f(y))=x.

Example of Inverse trigonometric functions: x= sin-1y

The list of inverse trigonometric functions with domain and range value is given below:

Functions

Domain

Range

Sin-1 x

[-1, 1] [-π/2, π/2]

Cos-1x

[-1, 1] [0, π/2]

Tan-1 x

R

(-π/2, π/2)

Cosec-1 x

R-(-1,1)

[-π/2, π/2]

Sec-1 x

R-(-1,1)

[0,π]-{ π/2}

Cot-1 x

R

[-π/2, π/2]-{0}

Inverse Trigonometric Formulas

To solve the different types of inverse trigonometric functions, inverse trigonometry formulas are derived from some basic properties of trigonometry. The inverse trigonometric formulas list is given below for reference to solve the problems.

S.No

Inverse Trigonometric Formulas

1

sin-1(-x) = -sin-1(x), x ∈ [-1, 1]

2

cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]

3

tan-1(-x) = -tan-1(x), x ∈ R

4

cosec-1(-x) = -cosec-1(x), |x| ≥ 1

5

sec-1(-x) = π -sec-1(x), |x| ≥ 1

6

cot-1(-x) = π – cot-1(x), x ∈ R

7

sin-1x + cos-1x = π/2 , x ∈ [-1, 1]

8

tan-1x + cot-1x = π/2 , x ∈ R

9

sec-1x + cosec-1x = π/2 ,|x| ≥ 1

10

sin-1(1/x) = cosec-1(x), if x ≥ 1 or x ≤ -1

11

cos-1(1/x) = sec-1(x), if x ≥ 1 or x ≤ -1

12

tan-1(1/x) = cot1(x), x > 0

13

tan-1 x + tan-1 y = tan-1((x+y)/(1-xy)), if the value xy < 1

14

tan-1 x – tan-1 y = tan-1((x-y)/(1+xy)), if the value xy > -1

15

2 tan-1 x = sin-1(2x/(1+x2)), |x| ≤ 1

16

2tan-1 x = cos-1((1-x2)/(1+x2)), x ≥ 0

17

2tan-1 x = tan-1(2x/(1-x2)), -1<x<1

18

3sin-1x = sin-1(3x-4x3)

19

3cos-1x = cos-1(4x3-3x)

20

3tan-1x = tan-1((3x-x3)/(1-3x2))

21

sin(sin-1(x)) = x, -1≤ x ≤1

22

cos(cos-1(x)) = x, -1≤ x ≤1

23

tan(tan-1(x)) = x, – ∞ < x < ∞.

24

cosec(cosec-1(x)) = x, – ∞ < x ≤ 1 or -1 ≤ x < ∞

25

sec(sec-1(x)) = x,- ∞ < x ≤ 1 or 1 ≤ x < ∞

26

cot(cot-1(x)) = x, – ∞ < x < ∞.

27

sin-1(sin θ) = θ, -π/2 ≤ θ ≤π/2

28

cos-1(cos θ) = θ, 0 ≤ θ ≤ π

29

tan-1(tan θ) = θ, -π/2 < θ < π/2

30

cosec-1(cosec θ) = θ, – π/2 ≤ θ < 0 or 0 < θ ≤ π/2

31

sec-1(sec θ) = θ, 0 ≤ θ ≤ π/2 or π/2< θ ≤ π

32

cot-1(cot θ) = θ, 0 < θ < π

33

\(\sin ^{-1}x +\sin ^{-1}y=\sin ^{-1}(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}), if x, y \geq 0 and x^{2}+y^{2} \leq 1\)

34

\(\sin ^{-1}x +\sin ^{-1}y=\pi -\sin ^{-1}(x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}})\), if x, y ≥ 0 and x2+y2>1.

35

\(\sin ^{-1}x +\sin ^{-1}y=\pi -\sin ^{-1}(x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}})\), if x, y ≥ 0 and x2+y2≤1.

36

\(\sin ^{-1}x +\sin ^{-1}y=\pi -\sin ^{-1}(x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}})\), if x, y ≥ 0 and x2 +y2>1.

37

\(\cos ^{-1}x +\cos ^{-1}y=\cos ^{-1}(xy-\sqrt{1-x^{2}}\sqrt{1-y^{2}})\), if x,

y >0 and x2+y2 ≤1.

38

\(\cos ^{-1}x +\cos ^{-1}y=\pi -\cos ^{-1}(xy-\sqrt{1-x^{2}}\sqrt{1-y^{2}})\), if x, y >0 and x2+y2>1.

39

\(\cos ^{-1}x +\cos ^{-1}y=\cos ^{-1}(xy+\sqrt{1-x^{2}}\sqrt{1-y^{2}})\), if x, y > 0 and x2+y2≤1.

40

\(\cos ^{-1}x +\cos ^{-1}y=\pi -\cos ^{-1}(xy+\sqrt{1-x^{2}}\sqrt{1-y^{2}})\),if x, y > 0 and x2 +y2>1.

The inverse trigonometric formulas list help the students to solve the problems in an easy way by applying those properties to find out the solutions.

For more information on inverse trigonometric formulas, visit BYJU’S – The Learning App and also watch interactive videos to learn the concepts in an easy and engaging way.


Practise This Question

In the figure shown, the two lines are parallel to each other. Which of the following options is incorrect?