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 17E
To determine

The values of six trigonometric functions using tanθ=2, sinθ<0 .

Expert Solution & Answer
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Answer to Problem 17E

  sinθ=−255, cosθ=−55,tanθ=2, cotθ=12,secθ=−5, cscθ=−52

Explanation of Solution

Given:

The values of expressions, tanθ=2, sinθ<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 tan is positive and sine is negative, so the angle should be in Quadrant III.

From the trigonometric ratios, we know that tan=OppositeAdjacent=21 .

So, using the Pythagorean theorem, it gives

  Hypotenus=(adjacent)2+(opposite)2=(1)2+(2)2=1+4=5

Since the angle is in third quadrant, so only tangent and cotangent will be positive, rest other trigonometric function will be negative.

So, using the reciprocal and quotient identities, it gives

  sinθ=OppositeHypotenus=−25=−255,cosθ=AdjacentHypotenus=−15=−55,cotθ=1tanθ=12,secθ=1cosθ=1−1/5=−5,cscθ=1sinθ=1−2/5=−52

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

  sinθ=−255, cosθ=−55,tanθ=2, cotθ=12,secθ=−5, cscθ=−52

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Chapter 5.1, Problem 16E
Next
Chapter 5.1, Problem 18E

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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