physicscatalyst.com logo




Properties of Inverse Trigonometric Functions





Table Of Contents


Properties of Inverse Trigonometric Functions

  • In last page,we cover about what are Inverse Trigonometric Functions and how they are used to find the angles corresponding to given trigonometric ratios, which is an essential concept in mathematics, physics, and engineering
  • In this article, we will discuss the properties of these inverse trigonometric functions.
  • It may be mentioned here that these results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined.
  • Some results may not be valid for all values of the domains of inverse trigonometric functions

Inverse of Negative x

$sin^{-1} (-x) = -sin^{-1} (x)$
$cos^{-1} (-x) = \pi - cos^{-1} (x)$
$tan^{-1} (-x) = -tan^{-1} (x)$
$sec^{-1} (-x) = \pi - sec^{-1} (x)$
$cosec^{-1} (-x) = -cosec^{-1} (x)$
$cot^{-1} (-x) = \pi - cot^{-1} (x)$
Lets see the proof of one of them
Proof on Sin
Let $sin^{-1} (-x) = y$
Then $-x = sin y$
or $x = - sin y$
or $x = sin (-y)$
Hence $sin^{-1}x = -y = - sin^{-1} (-x)$
Similary we can prove for others

Reciprocal of x

$sin^{-1} (\frac {1}{x}) = cosec^{-1} (x)$
$cos^{-1} (\frac {1}{x}) = sec^{-1} (x)$
$tan^{-1} (\frac {1}{x}) = cot^{-1} (x)$

Complementary Inverse function

$sin^{-1} (x) + cos^ {-1} (x) = \frac {\pi}{2}$
$sec^{-1} (x) + cosec^ {-1} (x) = \frac {\pi}{2}$
$tan^{-1} (x) + cot^ {-1} (x) = \frac {\pi}{2}$

Addition of Same Inverse function

$sin^{-1} (x) + sin^ {-1} (y) = sin ^{-1} (x \sqrt {1-y^2} + y \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$sin^{-1} (x) + sin^ {-1} (y) = \pi - sin ^{-1} (x \sqrt {1-y^2} + y \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$sin^{-1} (x) - sin^ {-1} (y) = sin ^{-1} (x \sqrt {1-y^2} - y \sqrt {1-x^2)}$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$sin^{-1} (x) - sin^ {-1} (y) = \pi - sin ^{-1} (x \sqrt {1-y^2} - y \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$cos^{-1} (x) + cos^ {-1} (y) = cos ^{-1} (x y - \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$cos^{-1} (x) + cos^ {-1} (y) = \pi - cos ^{-1} ((x y - \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$cos^{-1} (x) - cos^ {-1} (y) = cos ^{-1} (x y + \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$cos^{-1} (x) - cos^ {-1} (y) = \pi - cos ^{-1} (x y + \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$tan^{-1} (x) + tan^ {-1} (y)= tan^{-1} (\frac {x+y}{1-xy})$ , if $x,y > 0 $, $xy < 1$
$tan^{-1} (x) + tan^ {-1} (y)= \pi + tan^{-1} (\frac {x+y}{1-xy})$ , if $x,y > 0 $, $xy > 1$
$tan^{-1} (x) + tan^ {-1} (y)= tan^{-1} (\frac {x+y}{1-xy}) - \pi$ , if $x < 0, y > 0 $, $xy > 1$
$tan^{-1} (x) - tan^ {-1} (y)= tan^{-1} (\frac {x-y}{1+xy}) - \pi$ , if $xy > -1$
$tan^{-1} (x) + tan^ {-1} (y) + tan^ {-1} (z) = tan^{-1} (\frac {x+y+z - xyz}{1-xy-yz-xz})$

Double angle identity for Inverse functions

$ 2 sin^{-1} (x) = sin^{-1} (2x \sqrt {1-x^2})$ if $ -\frac {1}{\sqrt {2}} \leq x \frac {1}{\sqrt {2}} $
$ 2 cos^{-1} (x) = cos^{-1} (2x^2 -1)$
$2 tan^{-1} (x) = tan^{-1} (\frac {2x}{1-x^2})$ if $ -1 $2 tan^{-1} (x) = sin^{-1} (\frac {2x}{1+x^2})$ if $ |x| \leq 1$
$2 tan^{-1} (x) = cos^{-1} (\frac {1 -x^2}{1+x^2})$ if $ x \geq 0$
$3 sin^{-1} (x) = sin^{-1} (3x -4x^3)$
$3 cos^{-1} (x) = cos^{-1} (4x^3 - 3x)$
$3 tan^{-1} (x) = tan^{-1} (\frac {3x -x^3}{1-3x^2})$

Odd and Even Properties

Arcsine and arctangent are odd functions
$sin^{-1}(-x) = -sin^{-1} (x)$
$tan^{-1}(-x) = -tan^{-1}x$

Also Read




Go back to Class 12 Main Page using below links
Class 12 Maths Class 12 Physics Class 12 Chemistry Class 12 Biology


Latest Updates
Synthetic Fibres and Plastics Class 8 Practice questions

Class 8 science chapter 5 extra questions and Answers

Mass Calculator

3 Fraction calculator

Garbage in Garbage out Extra Questions7