# Evaluating inverse trigonometric functions

### Evaluating inverse trigonometric functions

#### Lessons

In this lesson, we will learn:

• Application of the Cancellation Laws
• Solving Expressions With One Inverse Trigonometry
• Evaluating Expressions With a Combination of Inverse and Non-Inverse Trigonometry
• Special Cases: Evaluating Functions With Numbers Outside of the Restrictions

Cancellation Laws:

$\sin^{-1} (\sin x) = x\;$, $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

$\sin (\sin^{-1} x) = x\;$, $-1 \leq x \leq 1$

$\cos^{-1} (\cos x) = x\;$, $0 \leq x \leq \pi$

$\cos (\cos^{-1} x) = x\;$, $-1 \leq x \leq 1$

$\tan^{-1} (\tan x) = x\;$, $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

$\tan (\tan^{-1} x) = x\;$, $-\infty$ < $x$ < $\infty$

Trigonometric Identity:

$\cos 2\theta = \cos^{2} \theta - \sin^{2} \theta$

• Introduction
Application of the Cancellation Laws

Introduction to Evaluating Inverse Trigonometric Functions

• 1.
Understanding the Use of Inverse Trigonometric Functions

Find the angles for each of the following diagrams.

a) b) • 2.

Find the angle for the following isosceles triangle. • 3.
Determining the Angles in Exact Values by Using Special Triangles

Find the angles for each of the following diagrams in exact value.

a) b) • 4.
Application of the Cancellation Laws

Solve the following inverse trigonometric functions:

a)
$\sin (\sin^{-1} 0.5)$

b)
$\cos^{-1} (\cos \frac{\pi}{4})$

c)
$\sin^{-1} (\sin \frac{3\pi}{4})$

• 5.
Solving Expressions With One Inverse Trigonometry

Solve the following inverse trigonometric functions:

a)
$\cos^{-1} \frac{1}{2}$

b)
$\sin^{-1} \frac{1}{2}$

• 6.
Evaluating Expressions With a Combination of Inverse and Non-Inverse Trigonometry

Solve the following inverse trigonometric functions:

a)
$\sin (\cos^{-1} \frac{\sqrt 3}{2})$

b)
$\cos (\sin^{-1} \frac{2}{3})$

c)
$\cos (2\tan^{-1} \sqrt 2)$

d)
$\cos (\sin^{-1} x)$

• 7.
Special Cases: Evaluating Functions With Numbers Outside of the Restrictions

Solve the following inverse trigonometric functions:

a)
$\cos^{-1} (\cos \frac{3\pi}{2})$

b)
$\sin^{-1} (\sin \frac{5\pi}{2})$