Precalculus with Limits
Precalculus with Limits
3rd Edition
ISBN: 9781133947202
Author: Ron Larson
Publisher: Cengage Learning
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Fundamentals of Trigonometric Identities
Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.
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Chapter 5.4, Problem 52E
To determine

Tofind:Thevalue of the trigonometric expression sec(v−u) for sinu=−725 and cosv=−45 where u and v are in quadrant III.

Expert Solution & Answer
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Answer to Problem 52E

Thevalue of the trigonometric expression sec(v−u) is 125117 .

Explanation of Solution

Given information:

The trigonometric expression is sec(v−u) . The value of sinu=−725 and cosv=−45 where u and v are in quadrant III.

Calculation:

Since, the value of u and v are in quadrant III. Let’s calculate the value of cosu and tanu in quadrant III using Pythagoras theorem.

  Precalculus with Limits, Chapter 5.4, Problem 52E , additional homework tip 1

Calculate the base of the triangle when sinu=−725 .

  Base=(25)2−(−7)2=576=±24

Since, the values lies in quadrant III. Take the negative value of base.

Calculate the value of cosu and tanu .

  cosu=−2425tanu=724

Similar, let’s calculate the value of sinv and tanv in quadrant III using Pythagoras theorem.

  Precalculus with Limits, Chapter 5.4, Problem 52E , additional homework tip 2

Calculate the base of the triangle when cosv=−45 .

  Base=(5)2−(−4)2=25−16=±3

Since, the values lies in quadrant III. Take the negative value of the base.

Calculate the value of sinv and tanv .

  sinv=−35tanv=34

Expand the given expression using sum formula.

  sec(v−u)=1cosu⋅cosv+sinu⋅sinv

Substitute the values in the above expression.

  sec(v−u)=1cosu⋅cosv+sinu⋅sinv=1(−2425)(−45)+(−725)(−35)=12596+21=125117

Therefore, thevalue of the trigonometric expression sec(v−u) is 125117 .

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Chapter 5.4, Problem 51E
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Chapter 5.4, Problem 53E

Chapter 5 Solutions

Precalculus with Limits

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