Answer sin θ = 0 , cos θ = − 1 , tan θ = 0 , cot θ = undefined , sec θ = − 1 , csc θ = undefined Explanation Given: The values of expressions, csc θ is undefined , cos θ < 0 Concept Used: Reciprocal identities of trigonometric functions: sin θ = 1 csc θ , csc θ = 1 sin θ , cos θ = 1 sec θ , sec θ = 1 cos θ , tan θ = 1 cot θ , cot θ = 1 tan θ Quotient Identities: tan θ = sin θ cos θ , cot θ = cos θ sin θ 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 csc π is undefined and cos π = − 1 < 0 . So, using the reciprocal and quotient identities, it gives sin θ = sin π = 0 , cos θ = cos π = − 1 , tan θ = sin θ cos θ = 0 − 1 = 0 , cot θ = 1 tan θ = 1 0 = undefined, sec θ = 1 cos θ = 1 − 1 = − 1 , csc θ = 1 sin θ = 1 0 = undefined, Thus, the values of the six trigonometric function is given by, sin θ = 0 , cos θ = − 1 , tan θ = 0 , cot θ = undefined , sec θ = − 1 , csc θ = undefined

Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Concept explainers

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

Chapter 5.1, Problem 19E

To determine

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

Expert Solution & Answer

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,

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

Chapter 5.1, Problem 18E

Chapter 5.1, Problem 20E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

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The values of six trigonometric functions using csc θ is undefined , cos θ &lt; 0 .