/*! 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 14 Evaluate the given quantities as... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 14

Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(0, \frac{\pi}{2}\right)\) and $$\cos u=\frac{1}{3} \quad\( and \)\quad \sin v=\frac{1}{4}$$ $$\cos (2 u)$$

Short Answer

Expert verified
The value of \(\cos (2u)\) is \(-\frac{7}{9}\).

Step by step solution

01

Write down the given values

We are given that: \[ \cos u = \frac{1}{3}, \quad \sin v = \frac{1}{4} \]
02

Use the double angle formula for cosine

The double angle formula for cosine is: \[ \cos (2u) = 2\cos^2{u} - 1 \]
03

Substitute the value of \(\cos u\) in the formula

We know that \(\cos u = \frac{1}{3}\). Now, let's substitute this value into the formula: \[ \cos (2u) = 2\left(\frac{1}{3}\right)^2 - 1 \]
04

Simplify the expression

By simplifying the expression, we get: \[ \cos (2u) = 2\left(\frac{1}{9}\right) - 1 \] \[ \cos (2u) = \frac{2}{9} - \frac{9}{9} \]
05

Calculate the value of \(\cos (2u)\)

Finally, let's calculate the value of \(\cos (2u)\): \[ \cos (2u) = -\frac{7}{9} \] So, the value of \(\cos (2u)\) is \(-\frac{7}{9}\).

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

Show (without using a calculator) that $$\sin 10^{\circ} \cos 20^{\circ}+\cos 10^{\circ} \sin 20^{\circ}=\frac{1}{2} \text { . }$$

Suppose \(-\frac{\pi}{2}<\theta<0\) and \(\cos \theta=0.3\). (a) Without using a double-angle formula, evaluate \(\cos (2 \theta)\) by first finding \(\theta\) using an inverse trigonometric function. (b) Without using an inverse trigonometric function, evaluate \(\cos (2 \theta)\) again by using a double-angle formula.

Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(-\frac{\pi}{2}, 0\right)\) and \(\tan u=-\frac{1}{7} \quad\) and \(\quad \tan v=-\frac{1}{8}\) $$\cos (2 u)$$

Find angles \(u\) and \(v\) such that \(\cos (2 u)=\cos (2 v)\) but \(\cos u \neq \cos v\).

Find a formula for \(\sin \left(\theta+\frac{\pi}{2}\right)\).
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

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.