/*! 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 8 Verify that the \(x\) -values ar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 8

Verify that the \(x\) -values are solutions of the equation. \(2 \cos ^{2} 4 x-1=0\) (a) \(x=\frac{\pi}{16}\) (b) \(x=\frac{3 \pi}{16}\)

Short Answer

Expert verified
Both x-values are indeed solutions of the equation, as both substitutions result in 0, which is the right side of the equation.

Step by step solution

01

Substitute the first given value into the equation

Substitute \(x=\frac{\pi}{16}\) into the equation \(2 \cos ^{2} 4 x-1=0\). It becomes \(2\cos^2{(4* \frac{\pi}{16})}-1= 2\cos^2{\frac{\pi}{4}}-1\). We know that \(\cos{\frac{\pi}{4}}\) equals \(\frac{\sqrt{2}}{2}\), so replacing \(\cos{\frac{\pi}{4}}\) by \(\frac{\sqrt{2}}{2}\) we get the equation \(2 * (\frac{\sqrt{2}}{2})^2 - 1 = 2 * \frac{1}{2} -1 = 1-1=0\).
02

Substitute the second given value into the equation

Now substitute \(x=\frac{3\pi}{16}\) into the equation \(2 \cos ^{2} 4 x-1=0\). It becomes \(2\cos^2{(4* \frac{3\pi}{16})}-1= 2\cos^2{\frac{3\pi}{4}}-1\). We know that \(\cos{\frac{3\pi}{4}}\) equals \(-\frac{\sqrt{2}}{2}\), so replacing \(\cos{\frac{3\pi}{4}}\) by \(-\frac{\sqrt{2}}{2}\) we get the equation \(2 * (-\frac{\sqrt{2}}{2})^2 - 1 = 2 * \frac{1}{2} - 1 = 1-1=0\).

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

Use the half-angle formulas to simplify the expression. $$\sqrt{\frac{1-\cos 6 x}{2}}$$

Find the exact value of the expression. $$\cos 120^{\circ} \cos 30^{\circ}+\sin 120^{\circ} \sin 30^{\circ}$$

Use the product-to-sum formulas to rewrite the product as a sum or difference. $$\sin 5 \theta \sin 3 \theta$$

Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos 8 x}{1+\cos 8 x}}$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin (x+\pi)-\sin x+1=0$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.