PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
8th Edition
ISBN: 9781337904285
Author: Larson
Publisher: CENGAGE L
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Question
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Chapter 5.5, Problem 92E
To determine

To find: The solutions of the given equation in the given interval. Then verify the solutions graphically.

Expert Solution & Answer
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Answer to Problem 92E

The solution of the given equation in the interval [0,2π) is x=0,π4,π2,3π4,π,5π4,3π2,7π4 .

Explanation of Solution

Given information:

The following equation:

  cos2x−cos6x=0

The following interval:

  [0,2π)

Formula used:

The following sum to product formula:

  cosu−cosv=−2sin(u+v2)sin(u−v2)

Calculation:

The given equation can be solved as follows:

  cos2x−cos6x=0                                      [Given]−2sin(6x+2x2)sin(6x−2x2)=0             [Sum to product formula]−2sin(8x2)sin(4x2)=0                             [Add and subtract]−2sin4xsin2x=0                                      [Simplify]sin4xsin2x=0                                           [Divide both sides by −2]sin4x=0          sin2x=0                            [Set each factor to 0]4x=arcsin(0)   2x=arcsin(0)                    [Inverse trigonometric function]4x=nπ             2x=nπ                             [Value of inverse]x=nπ4               x=nπ2                             [Simplify]x=0,π4,π2,3π4,π,5π4,3π2,7π4                  [Solutions in the interval [0,2π)]

The solution is verified graphically as:

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press Y= button.

Step 3: Enter the expression cos2x−cos6x and which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS), Chapter 5.5, Problem 92E

Interpretation:

In the interval [0,2π) , the value of the equation is 0 at x=0,π4,π2,3π4,π,5π4,3π2,7π4 . Hence, verified.

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Chapter 5.5, Problem 91E
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Chapter 5.5, Problem 93E

Chapter 5 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

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