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 13E
To determine

The values of six trigonometric functions using secϕ=−1715, sinϕ=817 .

Expert Solution & Answer
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Answer to Problem 13E

  sinϕ=817, cosϕ=−1517,tanϕ=−815, cotϕ=−158,secϕ=−1715, cscϕ=178

Explanation of Solution

Given:

The values of expressions, secϕ=−1715, sinϕ=817

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

  cosϕ=1secϕ=1−17/15=−1517,tanϕ=sinϕcosϕ=8/17−15/17=817⋅(−1715)=−815,cotϕ=1tanϕ=1−8/15=−158,cscϕ=1sinϕ=18/17=178

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

  sinϕ=817, cosϕ=−1517,tanϕ=−815, cotϕ=−158,secϕ=−1715, cscϕ=178

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Chapter 5.1, Problem 12E
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Chapter 5.1, Problem 14E

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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Publisher:Cengage Learning
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Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities; Author: Mathispower4u;https://www.youtube.com/watch?v=OmJ5fxyXrfg;License: Standard YouTube License, CC-BY