Chapters

Inverse Trigonometric Functions

NCERT Class 12 Mathematics — Chapter 2

Mathematics, in general, is fundamentally the science of self-evident things. — Felix Klein

2.1 Introduction

Trigonometric functions are not one-one and onto on their natural domains, so inverses do not exist unless we restrict domain and range. Inverse trigonometric functions are used heavily in calculus and engineering.

2.2 Basic Concepts

If ff is bijective, f−1f^{-1} exists and

f−1(f(x))=x,f(f−1(y))=yf^{-1}(f(x)) = x,\qquad f(f^{-1}(y)) = y

sin⁡−1x\sin^{-1}x is not (sin⁡x)−1(\sin x)^{-1}. In fact (sin⁡x)−1=1sin⁡x(\sin x)^{-1} = \dfrac{1}{\sin x}.

When no branch is mentioned, we mean the principal value branch.

Principal value branches

  • sin⁡−1:[−1,1]→[−π2,π2]\sin^{-1} : [-1,1] \to \left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]
  • cos⁡−1:[−1,1]→[0,π]\cos^{-1} : [-1,1] \to [0,\pi]
  • cosec⁡−1:R∖(−1,1)→[−π2,π2]∖{0}\operatorname{cosec}^{-1} : \mathbb{R}\setminus(-1,1) \to \left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\setminus\{0\}
  • sec⁡−1:R∖(−1,1)→[0,π]∖{π2}\sec^{-1} : \mathbb{R}\setminus(-1,1) \to [0,\pi]\setminus\left\{\dfrac{\pi}{2}\right\}
  • tan⁡−1:R→(−π2,π2)\tan^{-1} : \mathbb{R} \to \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)
  • cot⁡−1:R→(0,π)\cot^{-1} : \mathbb{R} \to (0,\pi)

Identities (on suitable domains)

sin⁡(sin⁡−1x)=x,sin⁡−1(sin⁡x)=x\sin(\sin^{-1}x)=x,\quad \sin^{-1}(\sin x)=x

(and similarly for the other functions, on their principal branches)

Example: sin⁡−112=π4\sin^{-1}\dfrac{1}{2}=\dfrac{\pi}{4}, because sin⁡π4=12\sin\dfrac{\pi}{4}=\dfrac{1}{2} and π4\dfrac{\pi}{4} lies in [−π2,π2]\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right].

2.3 Properties of Inverse Trigonometric Functions

Valid on principal branches, for suitable xx:

3sin⁡−1x=sin⁡−1(3x−4x3),x∈[−12,12]3\sin^{-1}x = \sin^{-1}(3x-4x^3),\qquad x\in\left[-\dfrac{1}{2},\dfrac{1}{2}\right]

3cos⁡−1x=cos⁡−1(4x3−3x),x∈[12,1]3\cos^{-1}x = \cos^{-1}(4x^3-3x),\qquad x\in\left[\dfrac{1}{2},1\right]

sin⁡−1(2x1−x2)=2sin⁡−1xor2cos⁡−1x\sin^{-1}(2x\sqrt{1-x^2}) = 2\sin^{-1}x \quad\text{or}\quad 2\cos^{-1}x

(depending on the interval for xx)

Note that sin⁡−1(sin⁡x)=x\sin^{-1}(\sin x)=x only when xx is in the principal range. For example

sin⁡−1(sin⁡3π5)=2π5\sin^{-1}\left(\sin\dfrac{3\pi}{5}\right)=\dfrac{2\pi}{5}

because 3π5\dfrac{3\pi}{5} is outside [−π2,π2]\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right].

Chapter Summary

  • Inverse trig functions need restricted domains
  • Principal values lie in the ranges listed above
  • sin⁡−1x≠(sin⁡x)−1\sin^{-1}x \mathrel{\htmlClass{course-neq}{\char"2260}} (\sin x)^{-1}
  • sin⁡(sin⁡−1x)=x\sin(\sin^{-1}x)=x on [−1,1][-1,1]; sin⁡−1(sin⁡x)=x\sin^{-1}(\sin x)=x on the principal interval

Exercises (NCERT)

  • Exercise 2.1 — principal values
  • Exercise 2.2 — properties and simplification
  • Miscellaneous Exercise — mixed problems