Defining the inverse trigonometric functions

Restricting the domains of sine, cosine and tangent, the definitions, domains, ranges and graphs of the inverse sine, cosine and tangent functions, exact values, calculator values in radians, and compositions such as sin of inverse cos.

The definitions

Each trigonometric function is restricted to an interval where it is one-to-one and takes every value once. The inverse function then swaps the domain and range.

Function	Domain	Range
$y = \sin^{-1}x$	$-1 \le x \le 1$	$-\frac{\pi}{2} \le y \le \frac{\pi}{2}$
$y = \cos^{-1}x$	$-1 \le x \le 1$	$0 \le y \le \pi$
$y = \tan^{-1}x$	all real $x$	$-\frac{\pi}{2} \lt y \lt \frac{\pi}{2}$

Inverse sine as a reflection

The solid curve is y = sin⁻¹x. It is the reflection in y = x of the part of y = sin x between −π/2 and π/2.

Try this

Why does the graph of $y = \sin^{-1}x$ stop at $x = \pm 1$?

Inverse cosine and inverse tangent

y = cos⁻¹x (solid) runs from π down to 0. y = tan⁻¹x (dashed) has horizontal asymptotes y = ±π/2.

Try this

At what point do the two curves cross?

Exact values

Example 1●

Find $\sin^{-1}\big(-\tfrac12\big)$, $\cos^{-1}\big(-\tfrac12\big)$ and $\tan^{-1}\sqrt3$.

$\sin^{-1}\big(-\tfrac12\big) = -\dfrac{\pi}{6}$, which is in $\big[-\frac{\pi}{2}, \frac{\pi}{2}\big]$.

$\cos^{-1}\big(-\tfrac12\big) = \dfrac{2\pi}{3}$, which is in $[0, \pi]$.

$\tan^{-1}\sqrt3 = \dfrac{\pi}{3}$.

Compositions

Example 2●●

Find the exact value of $\sin\big(\cos^{-1}\frac35\big)$.

Let $\theta = \cos^{-1}\frac35$, an acute angle with $\cos\theta = \frac35$.

In a right-angled triangle with adjacent side $3$ and hypotenuse $5$, the opposite side is $4$. So $\sin\theta = \frac45$.

Example 3: interpreting the restrictions●●●

Evaluate $\sin^{-1}\big(\sin\frac{5\pi}{6}\big)$.

$\sin\frac{5\pi}{6} = \frac12$.

$\sin^{-1}\frac12 = \frac{\pi}{6}$, not $\frac{5\pi}{6}$, because the answer must be in $\big[-\frac{\pi}{2}, \frac{\pi}{2}\big]$.

In general $\sin(\sin^{-1}x) = x$ for $-1 \le x \le 1$, but $\sin^{-1}(\sin x) = x$ only for $-\frac{\pi}{2} \le x \le \frac{\pi}{2}$.

Flashcards

Range of $sin^{-1}x$

$-\frac{\pi}{2} \le y \le \frac{\pi}{2}$

Range of $cos^{-1}x$

$0 \le y \le \pi$

Range of $ an^{-1}x$

$-\frac{\pi}{2} \lt y \lt \frac{\pi}{2}$

Domain of $sin^{-1}x$ and $cos^{-1}x$

$-1 \le x \le 1$

Practice

Find $\sin^{-1}\big(-\tfrac12\big)$. Type pi for $\pi$.
Find $\cos^{-1}\big(-\tfrac12\big)$.
Find $\tan^{-1}\sqrt3$.
Find $\sin\big(\cos^{-1}\frac35\big)$.
Find $\sin^{-1}\big(\sin\frac{5\pi}{6}\big)$.
Find $\cos\big(\cos^{-1} 0.3\big)$.
Find $\tan^{-1}(-1)$.
Which is undefined?
The range of $y = \cos^{-1}x$ is: