Answer sin x = 2 3 , cos x = − 5 3 , tan x = − 2 5 5 , cot x = − 5 2 , sec x = − 3 5 5 , csc x = 3 2 Explanation Given: The values of expressions, sin ( − x ) = − 2 3 , tan x = − 2 5 5 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. Observe that the sine function is an odd function, so sin ( − x ) = − 2 3 ⇒ − sin ( x ) = − 2 3 ⇒ sin x = 2 3 Now, using the reciprocal and quotient identities, it gives tan x = sin x cos x ⇒ cos x = sin x tan x = 2 / 3 − 2 5 / 5 = 2 3 ⋅ ( − 5 2 5 ) = − 5 3 , cot x = 1 tan x = 1 − 2 5 / 5 = − 5 2 5 = − 5 2 , sec x = 1 − 5 / 3 = − 3 5 = − 3 5 5 , csc x = 1 sin x = 1 2 / 3 = 3 2 Thus, the values of the six trigonometric function is given by, sin x = 2 3 , cos x = − 5 3 , tan x = − 2 5 5 , cot x = − 5 2 , sec x = − 3 5 5 , csc x = 3 2

Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Trigonometric Identities

Trigonometry in mathematics deals with the right-angled triangle’s angles and sides. By trigonometric identities, we mean the identities we use whenever we need to express the various trigonometric functions in terms of an equation.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse of normal trigonometric functions. Alternatively denoted as cyclometric or arcus functions, these inverse trigonometric functions exist to counter the basic trigonometric functions, such as sine (sin), cosi…

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Question

Chapter 5.1, Problem 15E

To determine

The values of six trigonometric functions using sin(−x)=−23, tanx=−255 .

Expert Solution & Answer

Answer to Problem 15E

sinx=23, cosx=−53,tanx=−255, cotx=−52,secx=−355, cscx=32

Explanation of Solution

Given:

The values of expressions, sin(−x)=−23, tanx=−255

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.

Observe that the sine function is an odd function, so

sin(−x)=−23⇒−sin(x)=−23⇒sinx=23

Now, using the reciprocal and quotient identities, it gives

tanx=sinxcosx⇒cosx=sinxtanx=2/3−25/5=23⋅(−525)=−53,cotx=1tanx=1−25/5=−525=−52,secx=1−5/3=−35=−355,cscx=1sinx=12/3=32

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

sinx=23, cosx=−53,tanx=−255, cotx=−52,secx=−355, cscx=32

Chapter 5.1, Problem 14E

Chapter 5.1, Problem 16E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

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Prob. 135RE Ch. 5 - Prob. 136RE Ch. 5 - Prob. 137RE Ch. 5 - Prob. 138RE Ch. 5 - Prob. 139RE Ch. 5 - Prob. 140RE Ch. 5 - Prob. 1CT Ch. 5 - Prob. 2CT Ch. 5 - Prob. 3CT Ch. 5 - Prob. 4CT Ch. 5 - Prob. 5CT Ch. 5 - Prob. 6CT Ch. 5 - Prob. 7CT Ch. 5 - Prob. 8CT Ch. 5 - Prob. 9CT Ch. 5 - Prob. 10CT Ch. 5 - Prob. 11CT Ch. 5 - Prob. 12CT Ch. 5 - Prob. 13CT Ch. 5 - Prob. 14CT Ch. 5 - Prob. 15CT Ch. 5 - Prob. 16CT Ch. 5 - Prob. 17CT Ch. 5 - Prob. 18CT Ch. 5 - Prob. 19CT Ch. 5 - Prob. 20CT Ch. 5 - Prob. 21CT Ch. 5 - Prob. 22CT Ch. 5 - Prob. 23CT Ch. 5 - Prob. 24CT

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The values of six trigonometric functions using sin ( − x ) = − 2 3 , tan x = − 2 5 5 .