Inverse Trigonometric Functions – Introduction, Graph, Derivatives and Examples

# Inverse Trigonometric Functions – Introduction, Graph, Derivatives and Examples

## Trigonometry Inverse

Inverse Trigonometric Functions – Introduction: In mathematics, the inverse of a function is a function that “undoes” the original function. If f is a function and x is an input, then f inverse (x) is the output that results from applying the inverse function to x. The inverse of a function can be found using the inverse function theorem.

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The inverse of a trigonometric function is another trigonometric function. The inverse of a trigonometric function can be found using the inverse function theorem.

There are three inverse trigonometric functions: inverse sine (arcsin), inverse cosine (arccos), and inverse tangent (arctan).

• The inverse sine function is denoted by the symbol “arcsin” and takes a real number as input and outputs the angle in radians that has a sine equal to the input.
• The inverse cosine function is denoted by the symbol “arccos” and takes a real number as input and outputs the angle in radians that has a cosine equal to the input.
• The inverse tangent function is denoted by the symbol “arctan” and takes a real number as input and outputs the angle in radians that has a tangent equal to the input.

## Introduction to Inverse Trig Function

• The inverse trigonometric functions are the inverse of the trigonometric functions. They are used to calculate angles when given a value for a trigonometric function.
• There are six inverse trigonometric functions: inverse sine (arcsin), inverse cosine (arccos), inverse tangent (arctan), inverse secant (arcsec), inverse cosecant (arccosec), and inverse Cotangent (arccot).
• Each inverse trigonometric function is defined by a particular equation. For example, the inverse sine function is defined by the equation y = arcsin x. This equation states that the inverse sine of a number x is the number y such that y = sin-1(x).

## Domain and Range

Domain is the set of all input values that a function can take, while the range is the set of all output values that a function can produce.Domain is the set of all input values that a function can take, while the range is the set of all output values that a function can produce.

## Inverse Trigonometric Functions Graphs

• The inverse trigonometric functions are arcsin, arccos, arctan, and arccot. These functions are all graphed on the same coordinate plane. The inverse trigonometric functions are all graphed on the same coordinate plane.
• The graphs of the inverse trigonometric functions are shown below. They are all graphed on the same coordinate plane.

## Graphs of all Inverse Circular Functions

Inverse circular functions are graphed on the same coordinate plane, but with the inverse function’s equation in place of the original function’s equation. Inverse circular functions are graphed by solving for x in the inverse function’s equation and then plotting the coordinates of the points that result.

## Inverse Trigonometric Functions Table

Inverse Trigonometric Functions

Function

Inverse

sin-1

arcsin

cos-1

arccos

tan-1

arctan

cot-1

arccot

## Inverse Trigonometric Functions Derivatives

• sin(x) = cos(x) · sec(x)
• cos(x) = sin(x) · csc(x)
• tan(x) = sec(x) · tan(x)
• cot(x) = csc(x) · cot(x)

## Solved example

A particle of mass 2 kg is projected horizontally with a speed of 10 m/s from a height of 10 m above the ground.

(a) What is the kinetic energy of the particle at the time of projection?

(b) What is the height of the particle at the time of impact?

(a) The kinetic energy of the particle at the time of projection is 100 J.

(b) The height of the particle at the time of impact is 5 m.

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