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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Fundamentals of Trigonometric Identities
Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.
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Chapter 5.4, Problem 89E

(a)

To determine

To represent:Thetrigonometric expression 12sin3θ+5cos3θ in the form of a2+b2sin(Bθ+C) and verify using graph.

(a)

Expert Solution
Check Mark

Answer to Problem 89E

The trigonometric expression 12sin3θ+5cos3θ can be represented as 13sin(3θ+0.3948) .

Explanation of Solution

Given information:

The given equationis 12sin3θ+5cos3θ .

Formula Used:

  a2+b2=r2

Proof:

Compare the expression 12sin3θ+5cos3θ with a2+b2sin(Bθ+C) .

  a=12,b=5,B=3

Calculate the value of a2+b2 .

  a2+b2=122+52=169=13

Calculate the value of C .

  C=tan−1(ba)=tan−1(512)=0.39479≈0.3948

Consider the left hand side of the equation.

  12sin3θ+5cos3θ=a2+b2sin(Bθ+C)=13sin(3θ+0.3948)

Draw the graph for 13sin(3θ+0.3948) and 12sin3θ+5cos3θ using graphical utility.

  PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS), Chapter 5.4, Problem 89E , additional homework tip 1

Figure (1)

From the above, it is clear that both the graph coincides. Therefore, the trigonometric expression 12sin3θ+5cos3θ can be represented as 13sin(3θ+0.3948) .

(b)

To determine

To represent:The trigonometric expression 12sin3θ+5cos3θ in the form of a2+b2cos(Bθ−C) and verify using graph.

(b)

Expert Solution
Check Mark

Answer to Problem 89E

The trigonometric expression 12sin3θ+5cos3θ can be represented as 13cos(3θ−1.1760) .

Explanation of Solution

Given information:

The given equationis 12sin3θ+5cos3θ .

Formula Used:

  a2+b2=r2

Proof:

Compare the expression 12sin3θ+5cos3θ with a2+b2cos(Bθ−C) .

  a=1,b=1,B=1

Calculate the value of a2+b2 .

  a2+b2=12+12=2

Calculate the value of C .

  C=tan−1(ab)=tan−1(125)=1.17601≈1.1760

Consider the left hand side of the equation.

  12sin3θ+5cos3θ=a2+b2cos(Bθ−C)=13cos(3θ−1.1760)

Draw the graph for 13cos(3θ−1.1760) and 12sin3θ+5cos3θ using graphical utility.

  PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS), Chapter 5.4, Problem 89E , additional homework tip 2

Figure (2)

From the above, it is clear that both the graph coincides. Therefore, the trigonometric expression 12sin3θ+5cos3θ can be represented as 13cos(3θ−1.1760) .

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Chapter 5.4, Problem 88E
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Chapter 5.4, Problem 90E

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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Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities; Author: Mathispower4u;https://www.youtube.com/watch?v=OmJ5fxyXrfg;License: Standard YouTube License, CC-BY