/*! 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 42 Solve the multiple-angle equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 42

Solve the multiple-angle equation. $$\sec 4 x-2=0$$

Short Answer

Expert verified
The solutions to the equation are \( x = \frac{\pi}{12} + \frac{n\pi}{2} \) and \( x = \frac{5\pi}{12} + \frac{n\pi}{2} \) for all integers n.

Step by step solution

01

Express the Equation in Terms of Cosine

Secant is the reciprocal of cosine. The equation \(\sec 4x - 2 = 0\) can be rewritten as \( \frac{1}{\cos 4x} - 2 = 0\)
02

Solve for Cosine

Solve the equation for \( \cos 4x \). This can be done by adding 2 to both sides, and taking the reciprocal of the obtained equation. This gives \( \cos 4x = \frac{1}{2} \)
03

Solve for the Multiple Angle

Use the cosine function to find the value of the multiple angle. The angles whose cosine equals half are \( \frac{\pi}{3} \) and \( \frac{5\pi}{3} \) and because cosine has period \( 2\pi \), there will also be solutions every \( 2\pi \) in addition to these angles. Thus, the solutions are \( 4x = \frac{\pi}{3} + 2n\pi \) and \( 4x = \frac{5\pi}{3} + 2n\pi \), with n any integer.
04

Solve for x

Lastly, divide all terms by 4. This gives \( x = \frac{\pi}{12} + \frac{n\pi}{2} \) and \( x = \frac{5\pi}{12} + \frac{n\pi}{2} \), with n any integer.

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 power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{4} x \cos ^{2} x$$

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

Proof (a) Write a proof of the formula for \(\sin (u+v)\) (b) Write a proof of the formula for \(\sin (u-v)\)

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

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

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.