# Evaluating inverse trigonometric functions

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##### Introduction
###### Lessons
1. Application of the Cancellation Laws

Introduction to Evaluating Inverse Trigonometric Functions

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##### Examples
###### Lessons
1. Understanding the Use of Inverse Trigonometric Functions

Find the angles for each of the following diagrams.

2. Find the angle for the following isosceles triangle.

1. Determining the Angles in Exact Values by Using Special Triangles

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

2. Application of the Cancellation Laws

Solve the following inverse trigonometric functions:

1. $\sin (\sin^{-1} 0.5)$
2. $\cos^{-1} (\cos \frac{\pi}{4})$
3. $\sin^{-1} (\sin \frac{3\pi}{4})$
3. Solving Expressions With One Inverse Trigonometry

Solve the following inverse trigonometric functions:

1. $\cos^{-1} \frac{1}{2}$
2. $\sin^{-1} \frac{1}{2}$
4. Evaluating Expressions With a Combination of Inverse and Non-Inverse Trigonometry

Solve the following inverse trigonometric functions:

1. $\sin (\cos^{-1} \frac{\sqrt 3}{2})$
2. $\cos (\sin^{-1} \frac{2}{3})$
3. $\cos (2\tan^{-1} \sqrt 2)$
4. $\cos (\sin^{-1} x)$
5. Special Cases: Evaluating Functions With Numbers Outside of the Restrictions

Solve the following inverse trigonometric functions:

1. $\cos^{-1} (\cos \frac{3\pi}{2})$
2. $\sin^{-1} (\sin \frac{5\pi}{2})$
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###### Topic Notes

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:

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