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 5.4, Problem 14E
To determine

To find: the exact values of the sine, cosine and tangent of the angle.

Expert Solution & Answer
Check Mark

Answer to Problem 14E

  sin165°=6+24,cos165°=−6+24,tan165°=3−2 .

Explanation of Solution

Given information:

Given angle,

  165°=135°+30°

Formula used:

  sin(u+v)=sinucosv+cosusinvcos(u+v)=cosucosv−sinusinvtan(u+v)=tanu+tanv1−tanutanv

Calculation:

Find the value of sine, cosine and tangent of the angle 165° by using the sum formulas as follows

  sin165°=sin(135°+30°)=sin135°cos30°+cos135°sin30°=(32)(12)+(12)(−12)=322−(122)=3−122=6+24

Find exact values of the cosine of the above angle as below

Taking cosine on both sides as below

  cos165°=cos(135°+30°)

Using the sum formula cos(u+v)=cosucosv−sinusinv , we get

  cos165°=cos(135°+30°)=cos135°cos30°−sin135°sin30°=(32)(−12)−(12)(12)=−(322)−122cos165°=−(322+122)=−(3+122)

Multiplying numerator and denominator by 2

  cos165°=−((3+1)2(22)2)=−(6+24)

Hence,

  cos165°=−(6+24) .

Find the exact values of the tangent of the above angle as below

Taking tangent on both sides as below

  tan165°=tan(135°+30°)=tan135°+tan30°1−tan135°tan30°=−1+131−(−1)×13=13−113+1tan165°=1−331+33=1−31+3

Hence,

  tan165°=1−31+3=(1−3)21−3=−4−232=3−2 .

Previous
Chapter 5.4, Problem 13E
Next
Chapter 5.4, Problem 15E

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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Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning
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