/*! 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 65 Use the half-angle formulas to s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 65

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

Short Answer

Expert verified
The simplified form of the given expression using half-angle formulas is \( \cos 3x \).

Step by step solution

01

Identify the Half-Angle Identity

The half-angle identity for cosine is given as \( \cos{\frac{\theta}{2}} = \sqrt{\frac{1 + \cos \theta}{2}} \). Notice the similarity of the given expression with the right-hand side of this identity.
02

Recognize the Angle

It's important to see that \( \theta \) in the half-angle identity is twice as large as the angle inside the cosine function of the given expression. In this case, \( \theta = 2(3x) = 6x \).
03

Apply the Half-Angle Identity

Apply the cosine half-angle identity to the expression and simplify. Given that the provided identity \( \cos{\frac{\theta}{2}} = \sqrt{\frac{1 + \cos \theta}{2}} \) resembles the given expression, one can simply write this as \( \cos{\frac{\theta}{2}} = \cos 3x \).

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 length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). $$r=21 \mathrm{cm}, \theta=35^{\circ}$$

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

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sec \frac{u}{2}=\pm \sqrt{\frac{2 \tan u}{\tan u+\sin u}}$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos 3 \beta=\cos ^{3} \beta-3 \sin ^{2} \beta \cos \beta$$

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$\left(\frac{1}{3}, \frac{2}{3}\right),\left(-1,-\frac{3}{2}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.