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 12E
To determine

To calculate:Thesolution of the equation sin2x+cosx=0 in the interval [0,2π) .

Expert Solution & Answer
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Answer to Problem 12E

The solution set of the equation sin2x+cosx=0 in the interval [0,2π) is {π2,3π2,7π6,11π6} .

Explanation of Solution

Given information:

The given equation is sin2x+cosx=0 .

Formula used:

Use the double angle formula,

  sin2x=2sinxcosx

Calculation:

Substitute the value of sin2x into given equation.

  sin2x+cosx=2sinxcosx+cosx=cosx(2sinx+1)

Further simplify.

  cosx(2sinx+1)=0

The solutions in the interval [0,2π) , substitute the factor cos(x)=0 .

  x=π2,3π2

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

Further simplify

  sinx=−12

Therefore the solution is

  x=7π6,11π6

The solution set is

  {π2,3π2}∪{7π6,11π6}={π2,3π2,7π6,11π6} .

Verify the solutions using graphical utility:

Draw the graph for the equation sin2x+cosx=0 as shown in Figure(1).

  Precalculus with Limits: A Graphing Approach, Chapter 5.5, Problem 12E

Figure(1)

From the above, it is clear that {π2,3π2,7π6,11π6} are the solutions.

Therefore, the solution of the equation sin2x+cosx=0 in the interval [0,2π){π2,3π2,7π6,11π6} .

Previous
Chapter 5.5, Problem 11E
Next
Chapter 5.5, Problem 13E

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