   Chapter 8.2, Problem 23E ### 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. 2 π 3

To determine

To calculate: The value of six trigonometric functions of an angle 2π3 without using a calculator.

Explanation

Given Information:

The angle is 2π3.

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

The formulas for negative angles are,

sin(θ)=sin(θ)cos(θ)=cos(θ)

The formula for double angles,

sin(θ)=2sin(θ)cos(θ)cos(θ)=2cos2(θ)1tan(θ)=sin(θ)cos(θ)

Calculation:

Consider the angle,

θ=2π3

Now, evaluate the value of six trigonometric functions for θ=2π3 using Trigonometric values of common angles.

sin(2π3)=2sin(π3)cos(π3)=2(32)(12)=(32)

Next,

cos(2π3)=2cos2(π3)1=2(14)1=12

Next,

tan(2π3)=sin(2π3)cos(2π3)=3

Evaluate the csc(2π3) using reciprocal identities

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