/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Evaluate \(\sin \left(\cos ^{-1}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 15

Evaluate \(\sin \left(\cos ^{-1} \frac{1}{3}\right)\)

Short Answer

Expert verified
The short version of the answer is: \(\sin \left(\cos ^{-1} \frac{1}{3}\right) = \frac{2\sqrt{2}}{3}\).

Step by step solution

01

Find the angle of \(\cos ^{-1} \frac{1}{3}\)

Since the exercise asks to find the sin value of the angle corresponding to \(\cos ^{-1} \frac{1}{3}\), let's denote this angle as a variable, say \(x\), where \(\cos x = \frac{1}{3}\).
02

Apply the Pythagorean Identity

We'll now use the Pythagorean identity, which states that \(\sin^2{x}+\cos^2{x}=1\) for any angle x. Since we know the value of \(\cos{x}\) is \(\frac{1}{3}\), we can find the value of \(\sin^2{x}\). Our equation becomes: \[\sin^2{x}+(\frac{1}{3})^2=1\]
03

Solve for \(\sin^2{x}\)

Now solve for \(\sin^2{x}\): \[\sin^2{x}=1-(\frac{1}{3})^2=1-\frac{1}{9}=\frac{8}{9}\]
04

Solve for \(\sin{x}\)

Take the square root of both sides to find the value of \(\sin{x}\): \[\sin{x}=\sqrt{\frac{8}{9}}=\frac{2\sqrt{2}}{3}\]
05

Final Answer

The final answer for \(\sin \left(\cos ^{-1} \frac{1}{3}\right)\) is \(\frac{2\sqrt{2}}{3}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate \(\sin ^{-1}\left(\sin \frac{2 \pi}{7}\right)\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with \(\tan u=-2\) and \(\tan v=-3\) Find exact expressions for the indicated quantities. $$ \tan (-u) $$

Show that $$ \tan \left(\theta+\frac{\pi}{2}\right)=-\frac{1}{\tan \theta} $$ for every angle \(\theta\) that is not an integer multiple of \(\frac{\pi}{2} .\) Interpret this result in terms of the characterization of the slopes of perpendicular lines.

Evaluate \(\tan \left(-\tan ^{-1} \frac{7}{11}\right)\)

Is cosine an even function, an odd function, or neither?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.