Inverse Trigonometric Functions - Varsity Tutors

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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 angleRemember 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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concept. Use space or enter to play video.concept thumbnailBeginner

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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 $\sin(30^\\circ)=\tfrac{1}{2}$.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

A$73.3^\\circ$B$1.28^\\circ$C$45^\\circ$D$60^\\circ$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

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.)Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$1.08$B$0.95$C$1.25$D$0.75$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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