Graphs and symmetries of inverse trigonometric functions

Transformations of the inverse trigonometric graphs with their new domains and ranges, the odd symmetry of inverse sine and inverse tangent, the point symmetry of inverse cosine, the identity sin⁻¹x + cos⁻¹x = π/2, and the graph of sin⁻¹(sin x).

Transformations

Work out the domain from the inside of the function, and the range from the outside.

y = a sin⁻¹(x/b)

a stretches the graph vertically and changes the range. b stretches it horizontally and changes the domain.

Try this

Which slider changes the domain?

Example 1●●

Find the domain and range of $y = 3\cos^{-1}(2x - 1)$.

Domain: $-1 \le 2x - 1 \le 1$, so $0 \le x \le 1$.

Range: $0 \le \cos^{-1}(2x - 1) \le \pi$, so $0 \le y \le 3\pi$.

Symmetries

Example 2●

Given $\cos^{-1}0.2 \approx 1.369$, find $\cos^{-1}(-0.2)$.

$\pi - 1.369 \approx 1.772$.

An important identity

Complementary angles

$$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \quad \text{for } -1 \le x \le 1$$

Example 3: proof●●●

Prove the identity.

Let $\theta = \sin^{-1}x$, so $\sin\theta = x$ with $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$.

Then $\cos\big(\frac{\pi}{2} - \theta\big) = \sin\theta = x$, and $0 \le \frac{\pi}{2} - \theta \le \pi$.

So $\frac{\pi}{2} - \theta$ is in the range of $\cos^{-1}$, and $\cos^{-1}x = \frac{\pi}{2} - \theta$. Rearranging gives the result.

The graph of sin⁻¹(sin x)

y = sin⁻¹(sin x)

It agrees with y = x only for −π/2 ≤ x ≤ π/2. Elsewhere it zigzags between −π/2 and π/2.

Try this

What is the period of this zigzag?

Flashcards

$sin^{-1}(-x)$

$-\sin^{-1}x$

$cos^{-1}(-x)$

$\pi - \cos^{-1}x$

$sin^{-1}x + cos^{-1}x$

$\frac{\pi}{2}$

Domain of $y = asin^{-1}rac{x}{b}$

$-b \le x \le b$

Practice

Find the domain of $y = 3\cos^{-1}(2x - 1)$. Enter the smallest and largest values of $x$ as min, max.
Find the range of the same function. Enter it as min, max. Type pi for $\pi$.
Given $\cos^{-1}0.2 \approx 1.369$, find $\cos^{-1}(-0.2)$, correct to three decimal places.
Find $\sin^{-1}0.7 + \cos^{-1}0.7$.
Find the domain of $y = \sin^{-1}\frac{x}{2}$. Enter it as min, max.
Find the range of $y = 2\sin^{-1}x$. Enter it as min, max.
Find $\sin^{-1}\big(\sin\frac{2\pi}{3}\big)$.
Which function is odd?
The horizontal asymptotes of $y = 2\tan^{-1}x$ are: