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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Concept explainers

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 5.2, Problem 85E

a.

To determine

To verify that the expression for s is equivalent to hcotθ .

a.

Expert Solution
Check Mark

Explanation of Solution

Given:

The length s of a shadow cast by a vertical gnomon of height h when the angle of the sun above the horizon is θ can be modeled by s=hsin(90°−θ)sinθ .

Concept used:

  • Cofunction Identities: The value of any trigonometric function of an angle θ is equal to the cofunction of the complement of θ . The cofunction identities given by

    •   sinθ=cos(π2−θ), sinθ=cos(π2−θ)tanθ=cot(π2−θ), cotθ=tan(π2−θ)secθ=csc(π2−θ), cscθ=sec(π2−θ)

  • Quotient Identities:

  tanθ=sinθcosθ, cotθ=cosθsinθ

Calculation:

In order to verify the expression for s , use the cofunction identities and simplify further as shown below,

  s=hsin(90°−θ)sinθs=hcosθsinθ (sin(90°−θ)=cosθ)s=h⋅cosθsinθs=h⋅cotθs=hcotθ

Thus, it verifies that the expression for s is equivalent to hcotθ .

b.

To determine

To complete the table when h=5 .

b.

Expert Solution
Check Mark

Answer to Problem 85E

    θ15°30°45°60°75°90°
    s 18.668.6652.891.340

Explanation of Solution

Given:

The length s of a shadow cast by a vertical gnomon of height h when the angle of the sun above the horizon is θ can be modeled by s=hsin(90°−θ)sinθ .

Calculation:

Using a graphing utility, the value of the s at different angles shown below,

    θ15°30°45°60°75°90°
    s 18.668.6652.891.340

c.

To determine

To verify that the expression for s is equivalent to hcotθ .

c.

Expert Solution
Check Mark

Answer to Problem 85E

  Maximum length at: 15°Minimum length at: 90°

Explanation of Solution

Given:

The length s of a shadow cast by a vertical gnomon of height h when the angle of the sun above the horizon is θ can be modeled by s=hsin(90°−θ)sinθ .

Calculation:

From the above table in part (b) it is clear that the maximum length of the shadow was at 15° , and the minimum length of the shadow was at 90° .

d.

To determine

The time of the day when the angle of the sun above the horizon is 90° .

d.

Expert Solution
Check Mark

Answer to Problem 85E

  Noon 12:00 PM

Explanation of Solution

Given:

The length s of a shadow cast by a vertical gnomon of height h when the angle of the sun above the horizon is θ can be modeled by s=hsin(90°−θ)sinθ .

Calculation:

Since the length of the shadow s was zero when the angle of the sun above the horizon was 90° . That means at that time the sun will be just at top of the gnomon. In a day the sub be at the top of head at the time of noon.

So, when the angle of the sun above the horizon is 90° , the time of the day would be approximately 12:00 PM noon.

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Chapter 5.2, Problem 84E
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Chapter 5.2, Problem 86E

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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Prob. 123CRCh. 5 - Prob. 124CRCh. 5 - Prob. 125CRCh. 5 - Prob. 126CRCh. 5 - Prob. 127CRCh. 5 - Prob. 128CRCh. 5 - Prob. 129CRCh. 5 - Prob. 130CRCh. 5 - Prob. 131CRCh. 5 - Prob. 132CRCh. 5 - Prob. 1CTCh. 5 - Prob. 2CTCh. 5 - Prob. 3CTCh. 5 - Prob. 4CTCh. 5 - Prob. 5CTCh. 5 - Prob. 6CTCh. 5 - Prob. 7CTCh. 5 - Prob. 8CTCh. 5 - Prob. 9CTCh. 5 - Prob. 10CTCh. 5 - Prob. 11CTCh. 5 - Prob. 12CTCh. 5 - Prob. 13CTCh. 5 - Prob. 14CTCh. 5 - Prob. 15CTCh. 5 - Prob. 16CTCh. 5 - Prob. 17CTCh. 5 - Prob. 18CTCh. 5 - Prob. 19CTCh. 5 - Prob. 20CTCh. 5 - Prob. 21CTCh. 5 - Prob. 22CTCh. 5 - Prob. 23CTCh. 5 - Prob. 24CTCh. 5 - Prob. 25CT
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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