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Inverse Trigonometric Functions

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Inverse Trigonometric Functions

Study Guide

Key Definition

Inverse trigonometric functions, such as $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$, are used to find the unknown measure of an angle in a right triangle when two side lengths are known.

Important Notes

The range of θ for $\sin^{-1}(x)$ is $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$ (domain: $-1 \leq x \leq 1$)
The range of θ for $\cos^{-1}(x)$ is $0 \leq \theta \leq \pi$ (domain: $-1 \leq x \leq 1$)
The range of θ for $\tan^{-1}(x)$ is $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$ (domain: all real numbers)
Inverse functions return angles in radians or degrees
Use a calculator set to the correct mode (degree or radian) when computing
Round answers to a sensible number of decimal places (e.g., one decimal place)

Mathematical Notation

$\sin^{-1}(x)$ is the inverse sine function

$\cos^{-1}(x)$ is the inverse cosine function

$\tan^{-1}(x)$ is the inverse tangent function

$\angle$ represents an angle

Remember to use proper notation when solving problems

Why It Works

Inverse trigonometric functions allow us to determine angles based on known side ratios, reversing the process of traditional trigonometric functions.

Remember

Inverse functions can be used to find angles in right triangles using side lengths. For example, $\tan^{-1}(\frac{opposite}{adjacent})$ gives the angle.

Quick Reference

Angle from sides:$\tan^{-1}(\frac{opposite}{adjacent})$

Angle from opposite and hypotenuse:$\sin^{-1}(\frac{opposite}{hypotenuse})$

Angle from adjacent and hypotenuse:$\cos^{-1}(\frac{adjacent}{hypotenuse})$

Understanding Inverse Trigonometric Functions

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Video explanation of this concept

Beginner

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Beginner Explanation

Inverse trigonometric functions return the angle when you know the ratio of sides in a right triangle. For example, if $\frac{opposite}{hypotenuse} = \tfrac{1}{2}$, then $\sin^{-1}(\tfrac{1}{2}) = 30^\\circ$ because $\sin(30^\\circ)=\tfrac{1}{2}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\tan^{-1}(\frac{10}{3})$ in degrees?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is 4 feet high with a slanted length of 10 feet (the hypotenuse). What is the angle of elevation above the horizontal? (Use degree mode.)

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the angle in a triangle where the opposite side is 7 units and the adjacent side is 24 units.

Challenge Quiz

Single Choice Quiz

Advanced

Calculate $\tan^{-1}(\frac{15}{8})$ in radians.

Recap

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Review key concepts and takeaways