Precalculus with Limits: A Graphing Approach
Precalculus with Limits: A Graphing Approach
7th Edition
ISBN: 9781305071711
Author: Ron Larson
Publisher: Brooks/Cole
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Chapter 5.5, Problem 15E
To determine

To calculate:Thesolution of the equation cos2x−cosx=0 in the interval [0,2π) .

Expert Solution & Answer
Check Mark

Answer to Problem 15E

The solution set of the equation cos2x−cosx=0 in the interval [0,2π) is {0,2π3,4π3} .

Explanation of Solution

Given information:

The given equation is cos2x−cosx=0 .

Formula used:

Use the sum to product formula.

  cosu−cosv=−2sin(u+v2)sin(u−v2)

Calculation:

Substitute the values of given cosine anglesinto sum to product formula.

  cos2x−cosx=−2sin(2x+x2)sin(2x−x2)=−2sin(3x2)sin(x2)

Further simplify.

  −2sin(3x2)sin(x2)=0

The solutions in the interval [0,2π) , substitute the factor sin(3x2)=0 .

  x=0,2π3,4π3

The solutions in the interval [0,2π) , substitute the factor sinx2=0 .

  x=0

The solution set is

  {0,2π3,4π3}∪{0}={0,2π3,4π3} .

Verify the solutions using graphical utility:

Draw the graph for the equation cos2x−cosx=0 as shown in Figure(1).

  Precalculus with Limits: A Graphing Approach, Chapter 5.5, Problem 15E

Figure(1)

From the above, it is clear that {0,2π3,4π3} are the solutions.

Therefore, the solution of the equation cos2x−cosx=0 in the interval [0,2π){0,2π3,4π3} .

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Chapter 5.5, Problem 14E
Next
Chapter 5.5, Problem 16E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 5.1 - Prob. 1ECh. 5.1 - Prob. 2ECh. 5.1 - Prob. 3ECh. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - Prob. 9ECh. 5.1 - Prob. 10E
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Find the solutions to a trig equation between 0 and 2pi; Author: Brian McLogan;https://www.youtube.com/watch?v=h7trDHjKCYc;License: Standard YouTube License, CC-BY