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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Question
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Chapter 5.5, Problem 55E
To determine

To find:The exact values of the sine, cosine and tangent at angle π8 using half-angle formula.

Expert Solution & Answer
Check Mark

Answer to Problem 55E

The exact values of the sine, cosine and tangent at angle π8 are 2−22 , 2+22 and 2−1 , respectively.

Explanation of Solution

Given information:

The given angle is π8 .

Calculation:

Since, angle π8 lies in Quadrant I. So, the values of sine, cosine and tangent will be positive.

The expression sinπ8 can be rewritten as,

  sin(π8)=sin(12⋅π4)

Write the expression to calculate the value of sin(12⋅π4) .

  sin(12⋅π4)=±12(1−cosπ4)=±12(1−(22)) {cosπ4=22}

Since, the value of sine is positive in Quadrant I. So, consider the positive value and simplify it.

  sin(12⋅π4)=2−22sinπ8=2−22

Write the expression to calculate the value of cosπ8 .

  cosπ8=cos(12⋅π4)=±1+cosπ42=±1+(22)2 {cosπ4=22}

Since, the value of cosine is positive in Quadrant I. So, consider the positive value and simplify it.

  cosπ8=1+(22)2=2+22×2=2+22

Similarly, write the expression to calculate the value of tanπ8 .

  tanπ8=tan(12⋅tanπ4)=±(1−cosπ4sinπ4)=±(1−(22)(12)) {cosπ4=22sinπ4=12}

Since, the value of tangent is positive in Quadrant I. So, consider the positive value and simplify it.

  tanπ8=1−(22)(12)=(2−22)(12)=2−1

Therefore, the exact values of the sine, cosine and tangent at angle π8 are 2−22 , 2+22 and 2−1 , respectively.

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Chapter 5.5, Problem 54E
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Chapter 5.5, Problem 56E

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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How to apply the law of sines to find the remaining parts of a triangle; Author: Brian McLogan;https://www.youtube.com/watch?v=NdRF18HWkmE;License: Standard YouTube License, CC-BY