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 20E
To determine

To calculate:Thesolution of the equation tan2x−2cosx=0 in the interval [0,2π) .

Expert Solution & Answer
Check Mark

Answer to Problem 20E

The solution set of the equation tan2x−2cosx=0 in the interval [0,2π) is {π6,5π6,3π2} .

Explanation of Solution

Given information:

The given equation is tan2x−2cosx=0 .

Formula used:

Use the double angle formula.

  sin2x=2sinxcosx

Calculation:

Simplify the given equation.

  tan2x−2cosx=0tan2x=2cosxsin2xcos2x=2cosxsin2x=2cosxcos2x .

Substitute the value of sin2x in the above equation.

  2sinxcosx=2cosxcos2xsinx=cos2xsinx=1−2sin2x

Further simplify,

  2sin2x+sinx−1=02sin2x+2sinx−sinx−1=0(2sinx−1)(sinx+1)=0

The solutions in the interval [0,2π) , substitute the factor 2sinx−1=0 .

  x=π6,5π6

The solutions in the interval [0,2π) , substitute the factor sinx+1=0 .

  x=3π2

Therefore, the solution set is

  {π6,5π6}∪{3π2}={π6,5π6,3π2}

Verify the solutions using graphical utility:

Draw the graph for the equation tan2x−2cosx=0 as shown in Figure(1).

  Precalculus with Limits: A Graphing Approach, Chapter 5.5, Problem 20E

Figure(1)

From the above, it is clear that {π6,5π6,3π2} are the solutions.

Therefore, the solution of the equation tan2x−2cosx=0 in the interval [0,2π){π6,5π6,3π2} .

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Chapter 5.5, Problem 19E
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Chapter 5.5, Problem 21E

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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Publisher:Cengage Learning
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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