/*! 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 43 Solve the multiple-angle equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 43

Solve the multiple-angle equation. $$2 \cos \frac{x}{2}-\sqrt{2}=0$$

Short Answer

Expert verified
x = \(\frac{\pi}{2}\) + 4\(\pi\)k, x = -\(\frac{3\pi}{2}\) + 4\(\pi\)k

Step by step solution

01

Isolate the cosine term

First, isolate the cos term on one side. This can be done by adding \(\sqrt{2}\) on both sides: 2 cos(\(\frac{x}{2}\)) - \(\sqrt{2}\) + \(\sqrt{2}\) = 0 + \(\sqrt{2}\)Which simplifies to: 2 cos(\(\frac{x}{2}\)) = \(\sqrt{2}\)
02

Solve for the angle \(\frac{x}{2}\)

Now, to solve for the angle \(\frac{x}{2}\), divide both sides by 2: cos(\(\frac{x}{2}\)) = \(\frac{\sqrt{2}}{2}\) Now, refer to the unit circle where cosine \(\frac{\sqrt{2}}{2}\) lies. It is related to the angle \(\frac{\pi}{4}\), and \(-\frac{3\pi}{4}\) as cosine is positive in both first and fourth quadrants: \(\frac{x}{2}\) = \(\frac{\pi}{4}\) + 2\(\pi\)k or \(\frac{x}{2}\) = -\(\frac{3\pi}{4}\) + 2\(\pi\)k where k is any integer.
03

Solve for x

Finally solve for x by multiplying the equation through by 2: x = 2(\(\frac{\pi}{4}\) + 2\(\pi\)k) or x = 2(-\(\frac{3\pi}{4}\) + 2\(\pi\)k)This simplifies to: x = \(\frac{\pi}{2}\) + 4\(\pi\)k and x = -\(\frac{3\pi}{2}\) + 4\(\pi\)k.

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 all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\sin \frac{x}{2}+\cos x-1=0$$

Write the expression as the sine, cosine, or tangent of an angle. $$\cos 130^{\circ} \cos 40^{\circ}-\sin 130^{\circ} \sin 40^{\circ}$$

Determine whether the statement is true or false. Justify your answer. Complementary Angles If \(\phi\) and \(\theta\) are complementary angles, then show that (a) \(\sin (\phi-\theta)=\cos 2 \theta\) and \((\mathrm{b}) \cos (\phi-\theta)=\sin 2 \theta\).

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the same viewing window. Use the graphs to determine whether \(y_{1}=y_{2}\) Explain your reasoning. $$y_{1}=\cos (x+2), \quad y_{2}=\cos x+\cos 2$$

Simplify the expression algebraically and use a graphing utility to confirm your answer graphically. $$\cos \left(\frac{3 \pi}{2}-x\right)$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.