/*! 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 17 Solve the equation. $$4 \cos ^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 17

Solve the equation. $$4 \cos ^{2} x-1=0$$

Short Answer

Expert verified
The solutions to the equation are \(x=\frac{\pi}{3} + 2\pi k, -\frac{\pi}{3} + 2\pi k, \frac{2\pi}{3} + 2\pi k, -\frac{2\pi}{3} + 2\pi k\) where \(k\) is an integer

Step by step solution

01

Rewrite the equation

Rewrite the equation as \(4 \cos ^{2} x = 1\). In this step, we're simply moving the constant term to the right side of the equation.
02

Isolate cosine square function

Divide both sides by 4 to isolate \(\cos^{2} x\), which gives us \(\cos^{2} x = \frac{1}{4}\). We've done this to get the cosine function by itself and to simplify the equation for further operations.
03

Take Square Root

Take the square root of both sides of the equation to isolate the cosine function. This yields \(\cos x = \pm \frac{1}{2}\). The 卤 symbol indicates that there are two solutions to this equation: one for \(+\frac{1}{2}\) and the other for \(-\frac{1}{2}\).
04

Calculate the angles

From trigonometric knowledge, it's known that \(\cos x = \frac{1}{2}\) when \(x=\frac{\pi}{3} + 2\pi k\) or \(x=-\frac{\pi}{3} + 2\pi k\) , and \(\cos x = -\frac{1}{2}\) when \(x= \frac{2\pi}{3} + 2\pi k\) or \(x= -\frac{2\pi}{3} + 2\pi k\). Here \(k\) is an integer which represents the multiple solutions due to the periodic nature of the cosine function.

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

Find the exact value of the expression. $$\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ}$$

Prove the identity. $$\sin (x+y)+\sin (x-y)=2 \sin x \cos y$$

Verify the identity. $$\frac{\sin x \pm \sin y}{\cos x+\cos y}=\tan \frac{x \pm y}{2}$$

Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\sin 6 x+\sin 2 x=0$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos \left(x+\frac{\pi}{4}\right)-\cos \left(x-\frac{\pi}{4}\right)=1$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.