Inverse Trigonometric Functions

Trigonometry Inverse

We already know about inverse operations. As we know addition and subtraction are inverse operations, and multiplication and division are inverse operations. Each operation has the opposite of its inverse. Similarly, inverse functions of the basic trigonometric functions are said to be inverse trigonometric functions. It also termed as arcus functions, anti trigonometric functions or cyclometric functions. 


The inverse of g is denoted by ‘g -1’.

Let y = f(y) = sin x, then its inverse is y = sin-1x.

In this article let us study the inverse of trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant functions. 


Image will be uploaded soon

Introduction to Inverse Trig Function

The inverse trigonometric functions actually perform the opposite operation of the trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. We know that trig functions are especially applicable to the right angle triangle. These six important functions are used to find the angle measure in a right triangle when two sides of the triangle measures are known.

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). 


  • sin-1x, cos-1x, tan-1x etc. denote angles or real numbers whose sine is x, cosine is x and tangent is x, provided that the answers given are numerically smallest available. These are also termed as arcsin x, arccosine x etc.

  • If there are two angles, one positive and the other negative having the same numerical value, then a positive angle should be taken.

  • Principal values, domains of inverse circular functions and range of inverse trig functions:


S. No.

Function

Domain

Range

1.

y = sin-1x

-1 ≤ x ≤ 1

-π/2 ≤ y ≤ π/2

2.

y = cos-1x

-1 ≤ x ≤ 1

0 ≤ y ≤ π

3.

y = tan-1x

x ∈ R

-π/2 < x < π/2

4.

y = cot-1x

x ∈ R

0 < y < π

5.

y = cosec-1x

x ≤ -1 or x ≥ 1

-π/2 ≤ y ≤ π/2, y ≠ 0

6.

y = sec-1x

x ≤ -1 or x ≥ 1

0 ≤ y ≤ π, y ≠ π/2


Inverse Trigonometric Functions Graphs

There are particularly six inverse trig functions for each trigonometric ratio. The inverse of six important trigonometric functions are:

  • Arcsine

  • Arccosine

  • Arctangent

  • Arccotangent

  • Arcsecant

  • Arccosecant


Graphs of all Inverse Circular Functions


1.Arcsine

 y = sin-1x, |x| ≤ 1, y ∈ [-π/2, π/2]

Image will be uploaded soon


  • sin-1x is bounded in [-π/2, π/2].

  • sin-1x is an increasing function.

  • In its domain, sin-1x attains its maximum value π/2 at x = 1 while its minimum value is -π/2 which occurs at x = -1.


 2.Arccosine

 y = cos-1x, |x| ≤ 1, y ∈ [0, π]

Image will be uploaded soon

  • cos-1x is bounded in [0, π].

  • cos-1x is a decreasing function.

  • In its domain, cos-1x attains its maximum value π at x = -1 while its minimum value is 0 which occurs at x = 1.

 

 3.Arctangent

 y = tan-1x,where  x ∈ R, y ∈ (- \[\frac{π}{2}\], \[\frac{π}{2}\])

Image will be uploaded soon

  • tan-1x is bounded in (-π/2, π/2).

  • tan-1x is an increasing function.

 

 4.Arccotangent

 y = cot-1x where x ∈ R, y ∈ (0, π)

Image will be uploaded soon

  • cot-1x is bounded in (0, π).

  •  cot-1x is a decreasing function.

 

 5.Arccosececant

 y = cosec-1x, |x| ≥ 1, y ∈ [-π/2, 0) ∪ (0, π/2].

Image will be uploaded soon

  • cosec-1x is bounded in [-π/2, π/2].

  • cosec-1x is a decreasing function.

  • In its domain, cosec-1x attains its maximum value π/2 at x = 1 while its minimum value is -π/2 which occurs at x = -1.

 

6. Arcsecant

y = sec-1x, |x| ≥ 1, y ∈ [0, π/2) ∪ (π/2, π].

Image will be uploaded soon

  1. sec-1x is bounded in [0, π].

  2.  sec-1x is an increasing function.

  3. In its domain,sec-1x attains its maximum value π at x = -1 while its minimum value is 0 which occurs at x = 1.

 

Here is the list of all the inverse trig functions with their notation, definition, domain and range of inverse trig functions

Inverse Trigonometric Functions Table

Function Name

Notation

Definition

Domain of  x

Range

Arcsine or inverse sine

y = sin-1(x)

x=sin y

−1 ≤ x ≤ 1

− π/2 ≤ y ≤ π/2

-90°≤ y ≤ 90°

Arccosine or inverse cosine

y=cos-1(x)

x=cos y

−1 ≤ x ≤ 1

0 ≤ y ≤ π

0° ≤ y ≤ 180°

Arctangent or

Inverse tangent

y=tan-1(x)

x=tan y

For all real numbers

− π/2 < y < π/2

-90°< y < 90°

Arccotangent or

Inverse Cot

y=cot-1(x)

x=cot y

For all real numbers

0 < y < π

0° < y < 180°

Arcsecant or

Inverse Secant

y = sec-1(x)

x=sec y

x ≤ −1 or 1 ≤ x

0≤y<π/2 or π/2<y≤π

0°≤y<90° or 90°<y≤180°

Arccosecant

y=cosec-1(x)

x=cosec y

x ≤ −1 or 1 ≤ x

−π/2≤y<0 or 0<y≤π/2

−90°≤y<0°or 0°<y≤90°


The derivatives of inverse trig functions are first-order derivatives. Let us check here the derivatives of all the six inverse functions.

Inverse Trigonometric Functions Derivatives

Inverse Trig Function

dy/dx

sin-1(x)

\[\frac{1}{\sqrt{(1 - x^{2})}}\]

cos-1(x)

\[\frac{-1}{\sqrt{(1 - x^{2})}}\]

tan-1(x)

\[\frac{1}{\sqrt{(1 + x^{2})}}\]

cot-1(x)

\[\frac{-1}{\sqrt{(1 + x^{2})}}\]

sec-1(x)

\[\frac{1}{[|x| \sqrt{(x^{2} - 1)]}}\]

cosec-1(x)

\[\frac{-1}{[|x| \sqrt{(x^{2} - 1)]}}\]


Solved Examples

Example 1: Find the value of tan-1(tan 9π/ 8 )

Solution:

 tan-1(tan9π/8)       

= tan-1tan ( π + π/8)

 = tan-1 (tan(π/ 8))

=π/ 8


Example 2: Find sin (cos-13/5).

Solution:

Suppose that, cos-1 3/5 = x

So, cos x = 3/5

We know, sin x =  \[\sqrt{1 - cos2x}\]

So, sin x = \[\sqrt{1 - 9/25}\] = 4/5

This implies, sin x = sin (cos-1 3/5) = 4/5

FAQ (Frequently Asked Questions)

1. What are the six trigonometric functions?

Trigonometry means the science of measuring triangles.Trigonometric functions can be simply defined as the functions of an angle of a triangle i.e. the relationship between the angles and sides of a triangle are given by these trig functions.

The six main trigonometric functions are as follows:

  • Sine (sin)

  • Cosine (cos)

  • Tangent (tan)

  • Secant (sec)

  • Cosecant (csc)

  • Cotangent (cot)

These functions are used to relate the angles of a triangle with the sides of that triangle where the triangle is the right angled triangle.. Trigonometric functions are important when studying triangles. To define these functions for the angle theta, begin with a right triangle. Each function relates the angle to two sides of a right triangle.