/*! 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 39 Solve the multiple-angle equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 39

Solve the multiple-angle equation. $$2 \cos 2 x-1=0$$

Short Answer

Expert verified
The solutions for x that satisfy the equation \(2 \cos 2x -1 = 0\) are \(x = 30^\circ, 150^\circ, -30^\circ, -150^\circ \).

Step by step solution

01

Interpret the Function

First, recognize that the problem is a trigonometric equation and involves cosine. The initial form, \(2 \cos 2x -1\), needs to be rearranged so the equation \(2 \cos 2x -1 = 0\) can be solved for \(x\).
02

Solve for Cosine

Add 1 to both sides to isolate the \(\cos 2x\) on one side of the equation. The equation becomes \(2 \cos 2x = 1\). Divide both sides by 2 to solve for \(\cos 2x\). Hence, \(\cos 2x = 0.5\).
03

Solve double-angle equation

Since the equation is in the format of a double angle, it needs to be solved accordingly. When cosine equals 0.5, the standard angles on the unit circle are \(60^\circ\) and \(300^\circ\). Therefore, \(2x = 60^\circ\) or \(2x = 300^\circ\). Remember, for cosine, we also have to consider angles on the negative side as the cosine function is positive in the first and fourth quadrants. As a result, \(2x = -60^\circ\) or \(2x = -300^\circ\). Thus, the solution to the double-angle equation is \(x = 30^\circ, 150^\circ, -30^\circ, -150^\circ \).

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

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\). $$\cos \left(x+\frac{\pi}{4}\right)+\cos \left(x-\frac{\pi}{4}\right)=1$$

Use the sum-to-product formulas to find the exact value of the expression. $$\sin 75^{\circ}+\sin 15^{\circ}$$

Simplify the expression algebraically and use a graphing utility to confirm your answer graphically. $$\tan (\pi+\theta)$$

Use the formulas given in Exercises 89 and 90 to write the trigonometric expression in the following forms.$$\text { (a) } \sqrt{a^{2}+b^{2}} \sin (B \theta+C)$$ $$\text { (b) } \sqrt{a^{2}+b^{2}} \cos (B \theta-C)$$ $$3 \sin 2 \theta+4 \cos 2 \theta$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

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.