Mathematics · Ch 4 — Inverse Trigonometric Functions

Principal Value of Inverse Trigonometric Functions

4.9

Principal Value of Inverse Trigonometric Functions

The principal value of an inverse trigonometric function at a point $x$ is the value of that inverse function at $x$ that lies inside its principal-value branch (range). E.g. the principal value of $\cos^{-1}\left(\tfrac{\sqrt3}2\right)$ is $\tfrac{\pi}6$ , since $\tfrac{\pi}6\in[0,\pi]$ .

Note

Tie-breaking convention. If two numerically equal but oppositely signed values both look like candidates, the principal value is taken to be the positive one (subject to lying in the correct range).

The full reference table (principal domain of the trig function → range of its inverse):

Function	Principal Domain	Range	Inverse Function	Domain	Range of Principal Value branch
sine	$\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$	$[-1,1]$	$\sin^{-1}$	$[-1,1]$	$\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$
cosine	$[0,\pi]$	$[-1,1]$	$\cos^{-1}$	$[-1,1]$	$[0,\pi]$
tangent	$\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$	$\mathbb{R}$	$\tan^{-1}$	$\mathbb{R}$	$\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$
cosecant	$\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]\setminus\{0\}$	$\mathbb{R}\setminus(-1,1)$	$\text{cosec}^{-1}$	$\mathbb{R}\setminus(-1,1)$	$\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]\setminus\{0\}$
secant	$[0,\pi]\setminus\left\{\tfrac{\pi}2\right\}$	$\mathbb{R}\setminus(-1,1)$	$\sec^{-1}$	$\mathbb{R}\setminus(-1,1)$	$[0,\pi]\setminus\left\{\tfrac{\pi}2\right\}$
cotangent	$(0,\pi)$	$\mathbb{R}$	$\cot^{-1}$	$\mathbb{R}$	$(0,\pi)$

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