Mathematics · Ch 2 — Inverse Trigonometric Functions

Definition of Inverse Trigonometric Functions

1. Definition of Inverse Trigonometric Functions

Recall (Class XI) that a function $f:X\to Y$ has an inverse $f^{-1}:Y\to X$ only when $f$ is bijective -- one-one (injective) and onto (surjective). If $f$ is bijective, its inverse is defined by

f^{-1}(y)=x \iff f(x)=y,

and the domain of $f^{-1}$ equals the range of $f$ , while the range of $f^{-1}$ equals the domain of $f$ .

Trigonometric functions are periodic, hence not one-one. Each of $\sin x,\cos x,\tan x,\cot x,\sec x,\text{cosec}\,x$ repeats its values infinitely often -- for instance $\sin 0=\sin\pi=\sin2\pi=0$ -- so, viewed on the whole of $\mathbb{R}$ (or wherever each is defined), none of them is one-one. A function that is not one-one has no inverse function in the ordinary sense: given $y=\sin x=0$ , there is no single $x$ that "the" inverse can return, since $x$ could be $0,\pi,2\pi,-\pi,\dots$ -- infinitely many angles all share the same sine.

Important

To obtain a genuine inverse function, the domain of the trigonometric function must first be restricted to a specific interval on which it becomes one-one, with its range unchanged (still onto its usual range). Different choices of interval are possible; each such restricted, bijective piece is called a branch of the inverse relation, and the value it returns is called a value of $\sin^{-1}x$ (or $\cos^{-1}x$ , etc.) on that branch. Exactly one branch, defined in the next section, is singled out by convention as the principal branch, and its values are the principal values -- these are what $\sin^{-1}x$ , $\cos^{-1}x$ , etc. mean whenever no other branch is stated.

General definition. If $y=\sin x$ is restricted to an interval on which sine is one-one and onto $[-1,1]$ , then for $x$ in that interval, $x=\sin^{-1}y$ means precisely: " $x$ is the angle, in the chosen branch, whose sine equals $y$ ." The same idea defines $\cos^{-1}x$ , $\tan^{-1}x$ , $\cot^{-1}x$ , $\sec^{-1}x$ and $\text{cosec}^{-1}x$ , once a branch is fixed for each.

Notation warning. The symbol $\sin^{-1}x$ (also written $\arcsin x$ ) denotes the inverse sine function. It does not mean $(\sin x)^{-1}=\dfrac{1}{\sin x}=\text{cosec}\,x$ . This is one of the most common notational confusions in the whole chapter, and the " $-1$ " in $\sin^{-1}x$ should always be read as "inverse of", never as an exponent.

Illustration. $\sin^{-1}\!\left(\dfrac12\right)=\dfrac{\pi}{6}$ means: $\dfrac{\pi}{6}$ is the angle, in the principal branch $\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$ , whose sine is $\dfrac12$ . Note that $\sin\left(\dfrac{5\pi}{6}\right)=\dfrac12$ as well, but $\dfrac{5\pi}{6}$ lies outside this branch, so it is never returned as the value of $\sin^{-1}\!\left(\dfrac12\right)$ .

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