# Inverse Trigonometric Functions

## 1.  What are trigonometric functions?

Trigonometric functions are a set of functions that relate the angles of right angled triangle to sides of right angled triangle. The Domain (inputs) to these functions are angles in degrees or radians and the Range (outputs) of these functions are real numbers.

Eg. y = sin(x) is a trigonometric function.

## 2. What are the different trigonometric functions?

The different trigonometric functions are: –

1. sin θ
2. cos θ
3. tan θ
4. cosec θ
5. sec θ
6. cot θ

## 3. Explain the different trigonometric functions.

a. sin θ: – If we consider a right angled triangle, the ratio of side opposite to the angle to its hypotenuse is called sin θ.

sin θ = $\mathrm{\frac{opposite}{hypotenuse}}$

b. cos θ: – If we consider a right angled triangle, the ratio of side adjacent to the angle to its hypotenuse is called cos θ.

cos θ = $\mathrm{\frac{adjacent}{hypotenuse}}$

c. tan θ: – If we consider a right angled triangle, the ratio of side opposite to the angle to the side adjacent to the angle is called tan θ.

tan θ = $\mathrm{\frac{opposite}{adjacent}}$

d. cosec θ: – If we consider a right angled triangle, the ratio of its hypotenuse to side opposite to the angle is called cosec θ.

It is the reciprocal of sin θ

cosec θ =$\mathrm{\frac{hypotenuse}{opposite}}$

e. sec θ: – If we consider a right angled triangle, the ratio of its hypotenuse to side adjacent to the angle is called sec θ

It is the reciprocal of cos θ

sec θ = $\mathrm{\frac{hypotenuse}{adjacent}}$

f. cot θ: – If we consider a right angled triangle, the ratio of side adjacent to the angle to the side opposite to the angle is called cot   θ. It is the reciprocal of tan θ

cot θ = $\mathrm{\frac{adjacent}{opposite}}$

## 4. What are inverse trigonometric functions?

Inverse Trigonometric Functions are those functions that relate the sides of right angled triangle to its angles. The Domain (inputs) to these functions are real numbers and the Range (output) is angles in degrees or radians.

Eg. y = sin -1(x)

In short inverse trigonometric functions are the inverse of trigonometric functions.

## 5. Give a chart showing the relation between the angles and the trigonometric value

 θ 0 30 45 60 90 sin θ 0 $\frac{1}{2}$ $\frac{1}{\sqrt 2}$ $\frac{\sqrt 3}{2}$ 1 cos θ 1 $\frac{\sqrt 3}{2}$ $\frac{1}{\sqrt 2}$ $\frac{1}{2}$ 0 tan θ 0 $\frac{1}{\sqrt 3}$ 1 ${\sqrt 3}$ Not defined cot θ Not defined ${\sqrt 3}$ 1 $\frac{1}{\sqrt 3}$ 0 sec θ 1 $\frac{2}{\sqrt 3}$ ${\sqrt 2}$ 2 Not defined cosec θ Not defined 2 ${\sqrt 2}$ $\frac{2}{\sqrt 3}$ 1

## 6. Specify the range and domain of different inverse trigonometric functions.

Domain and range of inverse functions:

 Inverse trig. functions Domain Range sin-1 θ [-1, 1] [-$\frac{\pi}{2}$$\frac{\pi}{2}$] cos-1 θ [-1, 1] [0 – $\pi$] tan-1 θ All Real (-$\frac{\pi}{2}$$\frac{\pi}{2}$) sec-1 θ All R-(-1, 1) [0 – $\pi$]-{$\frac{\pi}{2}$} cosec-1 θ All R-(-1, 1) [-$\frac{\pi}{2}$$\frac{\pi}{2}$]-{0} cot-1 θ All Real (0 – $\pi$)

## Important Formulae :

1. sin-1 $\left(\mathrm{\frac{1}{x}}\right)$ = cosec-1 x,   x $\geq$ 1 or x $\leq$ -1

2.  cos-1 $\left(\mathrm{\frac{1}{x}}\right)$ = sec-1 x,   x $\geq$ 1 or x $\leq$ -1

3.  tan-1 $\left(\mathrm{\frac{1}{x}}\right)$ = cot-1 x,   x $\in$ R

4.  cos-1 (-x) = $\pi$ – cos-1 (x),   x $\in$ [ -1 , 1 ]

5.  sec-1 (-x) = $\pi$ – sec-1 (x),   | x | $\leq$ 1

6.  cot-1 (-x) = $\pi$ – cot-1 (x),   x  $\in$ R

7.  sin-1 (-x) = – sin-1 (x),   x  $\in$ [ -1 , 1]

8.  tan-1 (-x) = – tan-1 (x),   x  $\in$ R

9.  cosec-1 (-x) = – cosec-1 (x),   | x |  $\geq$ 1

10.  sin-1 (x) + cos-1 (x) =   $\frac{\pi}{2}$ x

11.  tan-1 (x) + cot-1 (x) =   $\frac{\pi}{2}$ x

12.  cosec-1 (x) + sec-1 (x) =   $\frac{\pi}{2}$ |x| $\geq$ 1

13.  tan-1 (x) + tan-1 (y) =  tan-1 $\left(\frac{ x + y }{ 1 – xy }\right)$ xy $\lt$ 1

14.  tan-1 (x) – tan-1 (y) =  tan-1 $\left(\frac{ x – y }{ 1 + xy }\right)$ xy $\gt$ -1

15.  tan-1 x = sin-1  $\left(\frac{2x}{1 + x^2}\right)$

16.  tan-1 x = cos-1  $\left(\frac{1 – x^2}{1 + x^2}\right)$

17.  tan-1 x = tan-1  $\left(\frac{2x}{1 – x^2}\right)$

18.  sin-1  (2x $\sqrt{1 – x^2}$) = 2sin-1  x

19.  sin-1  (2x $\sqrt{1 – x^2}$) = 2cos-1  x

20.  sin-1  x = sin-1  (3x – 4$x^3$)

21.  cos-1  x = cos-1  (4$x^3$ – 3x)