   Chapter 8.2, Problem 66E ### 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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# Solving a Trigonometric Equation In Exercises 61-70, solve the equation for θ . Assume 0 ≤ θ ≤ 2 π . For some of the equations, you should use the trigonometric identities listed in this section. Use a spreadsheet or a graphing utility to verify your results. See Example 7. cos 2 θ = − 2 − 3 cos θ

To determine

To calculate: The values of θ for the trigonometric equation cos2θ=23cosθ and verify the result by graphing utility tool.

Explanation

Given Information:

The provided trigonometric equation is, cos2θ=23cosθ.

Formula used:

Double angle formula for trigonometric identity:

cos2θ=2cos2θ1

Calculation:

Consider the trigonometric equation, 2cos2θ1=23cosθ.

Simplify the provided trigonometry equation,

2cos2θ1=23cosθ2cos2θ+3cosθ1+2=02cos2θ+3cosθ+1=02cos2θ+2cosθ+cosθ+1=0

Further solve,

2cosθ(cosθ+1)+1(cosθ+1)=0(cosθ+1)(2cosθ+1)=0

Further solve,

cosθ+1=0cosθ=1

And,

2cosθ+1=02cosθ=1cosθ=12

Since, cosine is negative in second and third quadrants,

Thus, there are three solutions for trigonometric equation cos2θ=23cosθ for 0θ2π, which can be mathematically determined as:

First value:

cosθ=1cosθ=cosπθ=π=3.141592654

Second value:

cosθ=12cosθ=cos2π3θ=2π3θ=2.094395102

Third value:

cosθ=12cosθ=cos4π3θ=4π3θ=4

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