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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Trigonometric Identities
Trigonometry in mathematics deals with the right-angled triangle’s angles and sides. By trigonometric identities, we mean the identities we use whenever we need to express the various trigonometric functions in terms of an equation.
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverse of normal trigonometric functions. Alternatively denoted as cyclometric or arcus functions, these inverse trigonometric functions exist to counter the basic trigonometric functions, such as sine (sin), cosi…
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Chapter 5.5, Problem 38E
To determine

To write:The expression cos8x in first power of cosine and verify both the expressions by graphing utility.

Expert Solution & Answer
Check Mark

Answer to Problem 38E

The expression cos8x can be written as 1128(35+28cos4x+cos8x+48cos2x+16cos2xcos4x) as a single power of cosine.

Explanation of Solution

Given information:

The expression is given as cos8x .

Formula used:

Use the power reducing formula cos2x=1+cos2x2 .

Calculation:

As per power reducing formula.

  cos2x=1+cos2x2

Substitute the value in the above expression.

  cos8x=(cos2x)4=(1+cos2x2)4=116(1+cos2x)2(1+cos2x)2=116(1+2cos2x+cos22x)(1+2cos2x+cos22x)

Further simplify.

  cos8x=116(1+2cos2x+1+cos4x2)(1+2cos2x+1+cos4x2)=116(1+2cos2x+1+cos2x2)(1+2cos2x+1+cos2x2)=116(3+4cos2x+cos4x2)(3+4cos2x+cos4x2)

Simplify further,

  cos8x=164(9+16cos22x+cos24x+24cos2x+8cos2xcos4x+6cos4x)=164(9+16(1+cos4x2)+(1+cos8x2)+24cos2x+8cos2xcos4x+6cos4x)=1128(35+28cos4x+cos8x+48cos2x+16cos2xcos4x)

Therefore the expression cos8x can be written as 1128(35+28cos4x+cos8x+48cos2x+16cos2xcos4x) as a single power of cosine.

Verify both the expressions using graphical utility:

Draw the graph for the expression cos8x as shown in the Figure (1).

  Precalculus with Limits: A Graphing Approach, Chapter 5.5, Problem 38E , additional homework tip 1

Figure (1)

Draw the graph for the expression 1128(35+28cos4x+cos8x+48cos2x+16cos2xcos4x) as shown in the Figure (2).

  Precalculus with Limits: A Graphing Approach, Chapter 5.5, Problem 38E , additional homework tip 2

Figure (2)

As both the expressions shows same values in Figure (1) and Figure (2) at each point.

Therefore both the expressions cos8x and 1128(35+28cos4x+cos8x+48cos2x+16cos2xcos4x) are same.

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Chapter 5.5, Problem 37E
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Chapter 5.5, Problem 39E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

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Inverse Trigonometric Functions; Author: Professor Dave Explains;https://www.youtube.com/watch?v=YXWKpgmLgHk;License: Standard YouTube License, CC-BY