   Chapter 8.2, Problem 26E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Evaluating Trigonometric Functions In Exercises 19-32, evaluate the six trigonometric functions of the angle without using a calculator. See Examples 2 and 3. 4 π 3

To determine

To calculate: The value of six trigonometric functions of an angle 4π3 without using a calculator.

Explanation

Given Information:

The angle is 4π3.

Formula used:

Trigonometric values of Common angles:

 θ(degrees) 0° 30° 45° 60° 90° 180° 270° θ(radians) 0 π6 π4 π3 π2 π 3π2 sinθ 0 12 22 32 1 0 −1 cosθ 1 32 22 12 0 −1 0 tanθ 0 33 1 3 Undefined 0 Undefined

Sum and difference formulas:

sin(A±B)=sinAcosB±cosAsinBcos(A±B)=cosAcosBsinAsinBtan(A±B)=tanA±tanB1tanAtanB

Calculation:

Consider the angle,

θ=4π3

Write 4π3 as π+π3. Now, evaluate the six trigonometric function as follows,

sin(4π3)=sin(π+π3)

Apply the sum rule of Sine function,

sin(π+π3)=(sinπcosπ3+cosπsinπ3)

Solve the above function using Trigonometric value,

(sinπcosπ3+cosπsinπ3)=((0)(12)+(1)(32))=(032)=32

Now, evaluate the cosine function for θ=4π3.

cos(4π3)=cos(π+π3)

Apply the sum rule of cosine function,

cos(π+π3)=(cosπcosπ3sinπsinπ3)

Solve the above function using Trigonometric value,

(cosπcosπ3sinπsinπ3)=((1)(12)+(0)(32))=(120)=12

Now, evaluate the tangent function for θ=4π3

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