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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Question
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Chapter 5.1, Problem 19E
To determine

The values of six trigonometric functions using cscθ is undefined, cosθ<0 .

Expert Solution & Answer
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Answer to Problem 19E

  sinθ=0, cosθ=−1,tanθ=0, cotθ=undefined,secθ=−1, cscθ=undefined

Explanation of Solution

Given:

The values of expressions, cscθ is undefined, cosθ<0

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.
  • In Quadrant II, only sine and cosecant are positive.
  • In Quadrant III, only tangent and cotangent are positive.
  • In Quadrant 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.

Since cosecant (csc) is undefined and cosine is negative, so the angle must be π or 180° .

Because cscπ ​is undefined and cosπ=−1<0 .

So, using the reciprocal and quotient identities, it gives

  sinθ=sinπ=0,cosθ=cosπ=−1,tanθ=sinθcosθ=0−1=0,cotθ=1tanθ=10=undefined,secθ=1cosθ=1−1=−1,cscθ=1sinθ=10=undefined,

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

  sinθ=0, cosθ=−1,tanθ=0, cotθ=undefined,secθ=−1, cscθ=undefined

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Chapter 5.1, Problem 18E
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Chapter 5.1, Problem 20E

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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Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning
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