sin⁻¹, cos⁻¹, tan⁻¹ and principal values

5 minAS Level1.5.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can explain what $\sin^{-1}x$ , $\cos^{-1}x$ and $\tan^{-1}x$ mean, state the range of principal values for each, and evaluate them exactly for the special angles.

The undo button has a problem

Ask "which angle has sine $\tfrac12$ ?" and the honest answer is: loads. $30^\circ$ , $150^\circ$ , $390^\circ$ , $-210^\circ$ …

Sine isn't one-one (many inputs give the same output). So it can't have an inverse as it stands.

The fix is the same trick as in the inverse-functions lesson:

Restrict the domain to a piece of the graph that is one-one.
Invert just that piece.

The calculator's $\sin^{-1}$ button returns the one answer from that piece. It's called the principal value.

Definition

Principal value

$\sin^{-1}x$ , $\cos^{-1}x$ and $\tan^{-1}x$ denote the principal values: the single angle in a fixed range whose sine, cosine or tangent is $x$ .

Read $\sin^{-1}x$ as "the angle whose sine is $x$ ". (Calculators and some books call it arcsin.)

	Domain of $x$	Principal values (radians)	(degrees)
$\sin^{-1}x$	$-1 \leqslant x \leqslant 1$	$-\tfrac{\pi}{2} \leqslant \sin^{-1}x \leqslant \tfrac{\pi}{2}$	$-90^\circ$ to $90^\circ$
$\cos^{-1}x$	$-1 \leqslant x \leqslant 1$	$0 \leqslant \cos^{-1}x \leqslant \pi$	$0^\circ$ to $180^\circ$
$\tan^{-1}x$	all real $x$	$-\tfrac{\pi}{2} < \tan^{-1}x < \tfrac{\pi}{2}$	$-90^\circ$ to $90^\circ$ (not included)

Short-hand you'll see below: $[-1, 1]$ means "from $-1$ to $1$ , ends included". Round brackets, like $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$ , mean the ends are not included.

Which half of the circle?

Picture the unit circle:

$\sin^{-1}$ and $\tan^{-1}$ answer in the right half (quadrants 1 and 4): negative inputs give negative angles.
$\cos^{-1}$ answers in the top half (quadrants 1 and 2): negative inputs give obtuse angles.

Each half covers every possible value of that ratio exactly once. That's why these ranges were chosen.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.