Precalculus with Limits: A Graphing Approach
Precalculus with Limits: A Graphing Approach
7th Edition
ISBN: 9781305071711
Author: Ron Larson
Publisher: Brooks/Cole
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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 5.1, Problem 12E
To determine

The values of six trigonometric functions using cotϕ=−5, sinϕ=2626 .

Expert Solution & Answer
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Answer to Problem 12E

  sinϕ=2626, cosϕ=−52626,tanϕ=−15, cotϕ=−5,secϕ=−265, cscϕ=26

Explanation of Solution

Given:

The values of expressions, cotϕ=−5, sinϕ=2626

Concept Used:

  • Reciprocal identities of trigonometric functions:

    •   sinθ=1cscθ, cscθ=1sinθ,cosθ=1secθ, secθ=1cosθ,tanθ=1cotθ, cotθ=1tanθ

  • Quotient Identities:

    •   tanθ=sinθcosθ, cotθ=cosθsinθ

  • In Quadrant I, all trigonometric functions are positive.
  • InQuadrant II, only sine and cosecant are positive.
  • InQuadrant III, only tangent and cotangent are positive.
  • InQuadrant IV, only cosine and secant are positive.

Calculation:

In order tofind the values of the six trigonometric functions, use the reciprocal and quotient identities of trigonometric functions.

Using the reciprocal and quotient identities, it gives

  cotϕ=cosϕsinϕ⇒cosϕ=cotϕ⋅sinϕ=(−5)⋅(2626)=−52626,tanϕ=1cotϕ=1−5=−15,secϕ=1cosϕ=1−52626=−26526=−265,cscϕ=1sinϕ=126/26=2626=26

Thus, the values of the six trigonometric function is given by,

  sinϕ=2626, cosϕ=−52626,tanϕ=−15, cotϕ=−5,secϕ=−265, cscϕ=26

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Chapter 5.1, Problem 11E
Next
Chapter 5.1, Problem 13E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 5.1 - Prob. 1ECh. 5.1 - Prob. 2ECh. 5.1 - Prob. 3ECh. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - Prob. 9ECh. 5.1 - Prob. 10E
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Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning
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