# Inverse Trigonometric Functions

The definition of Trigonometry can be given through inverse Trigonometric functions.

## Trigonometry Basics:

Trigonometry basics include the basic trigonometry and trigonometric ratios such as $sin x , cos x , tanx , cosec x , sec x$  and $cot x$.

The following article from Byju’s discusses the basic definition of another tool of trigonometry – Inverse Trigonometric Functions.

## Inverse Trigonometric Functions:

Inverse trigonometric functions are also called “Arc Functions” since for a given value of a trigonometric function, they produce the length of arc needed to obtain that particular value.

### Arcsine Function:

Arcsine function is an inverse of the sine function denoted by sin-1x. It is represented in the graph as shown below:

$y = sin^{-1} x$(arcsinex)

Domain & Range of arcsine function:

$Domain: -1 \le x \le 1$

$Range: \frac {-\pi}{2} \le y \le \frac{\pi}{2}$

### Arccosine Function:

Arccosine function is the inverse of the cosine function denoted by cos-1x. It is represented in the graph as shown below:

$y = cos^{-1} x$(arcosinex)

Domain & Range:

$Domain: -1 \le x \le 1$

$Range: 0 \le y \le \pi$

### Arctangent Function:

Arctangent function is the inverse of the tangent function denoted by tan-1x. It is represented in the graph as shown below:

$y = tan^{-1} x$(arctangentx)

Domain & Range:

$Domain: – \infty < x < \infty$

$Range: \frac {-\pi}{2} < y < \frac{\pi}{2}$

### Arccotangent Function:

Arccotangent function is the inverse of the cotangent function denoted by cot-1x. It is represented in the graph as shown below:

$y = cot^{-1} x$ (arccotangentx)

Domain & Range:

$Domain: -\infty <x<\infty$

$Range: 0 < y < \pi$

### Arcsecant Function:

What is arcsecant (arcsecx)function?

Arcsecant function is the inverse of the secant function denoted by sec-1x. It is represented in the graph as shown below:

$y = sec^{-1} x$(arcsecantx)

Domain & Range:

$Domain: -\infty \le x \le -1 or 1 \le x \le \infty$

$Range: 0 \le y \le \pi , y \ne \frac{\pi}{2}$

### Arccosecant Function:

What is arccosecant (arccscx) function?

Arccosecant function is the inverse of the cosecant function denoted by cosec-1x. It is represented in the graph as shown below:

$y = cosec^{-1} x$(arccosecantx)

Domain & Range:

$Domain: -\infty \le x \le -1 or 1 \le x \le \infty$

$Range: – \frac{\pi}{2} < y < \frac{\pi}{2} y \ne 0$

Examples:Find the value of x, for sin(x) = 2.

Solution:sinx = 2

x =sin-1(2), which is not possible.

Hence there is no value of x for which sin x = 2; since the domain of $sin^{-1}x$  is -1 to 1 for the values of x.

Learn more about inverse trigonometric functions with the help of NCERT Solutions, refer to this link for NCERT Solution for Inverse Trigonometric Functions

#### Practise This Question

if a function f(x) is discontinuous in the interval (a,b) then baf(x)dx never exists.