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:

Arcsine Function

\(y = sin^{-1} x \)(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:

Arccosine Function\(y = cos^{-1} x \)(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:

Arctangent Function

\(y = tan^{-1} x \)(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:

Arccotangent Function

\(y = cot^{-1} x \) (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:

Arcsecant Function

\(y = sec^{-1} x \)(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:

Arccosecant Function

\(y = cosec^{-1} x \)(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.