/*! 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 48 Find the exact value of the trig... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 48

Find the exact value of the trigonometric expression given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\sin (u+v)$$

Short Answer

Expert verified
\(\sin(u+v)=\frac{128}{125}\)

Step by step solution

01

Identify and Apply the Formula

The formula for \(\sin(u+v)\) is \(\sin(u+v)=\sin u \cos v + \cos u \sin v\). This will be applied to the given values.
02

Find Missing Trigonometric Values

Since we are given \(\sin(u)\) and \(\cos(v)\), we'll need to find \(\cos(u)\) and \(\sin(v)\) using the Pythagorean identities. Remember, in quadrant III, sine and cosine are both negative. For \(\cos(u)\), we use the identity \(\cos^2(u)\)=1-\(\sin^2(u)\). Substituting the given \(\sin(u)=-\frac{7}{25}\) will give us \(\cos(u)\)=-\(\sqrt{1-\(-\frac{7}{25}\)^2}\) = -\(\sqrt{\frac{576}{625}}\) = -\(\frac{24}{25}\).Similarly, for \(\sin(v)=-\sqrt{1-\cos^2(v)\)=-\sqrt{1-\(-\frac{4}{5}\)^2}= -\(\frac{3}{5}\)
03

Substitute and Simplify the Expression

Substituting \(\sin(u) = -\frac{7}{25}\), \(\cos(u) = -\frac{24}{25}\), \(\cos(v)= -\frac{4}{5}\), and \(\sin(v)=-\frac{3}{5}\) into the formula will give us \(\sin(u+v)=\(-\frac{7}{25}\)*-\frac{4}{5} + -\frac{24}{25} * -\frac{3}{5}\) = \(\frac{56}{125} + \frac{72}{125} = \frac{128}{125}\). Hence \(\sin(u+v)\)= \(\frac{128}{125}\).

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

Simplify the expression algebraically and use a graphing utility to confirm your answer graphically. $$\sin \left(\frac{3 \pi}{2}+\theta\right)$$

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

Prove the identity. $$\cos \left(\frac{5 \pi}{4}-x\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x)$$

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

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\cos \left(\theta+\frac{\pi}{2}\right)-\cos \left(\theta-\frac{\pi}{2}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.