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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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(π2θ), sinθ=cos(π2θ)tanθ=cot(π2θ), cotθ=tan(π2θ)secθ=csc(π2θ), cscθ=sec(π2θ)

  • 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=hcosθsinθs=hcotθs=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.

Chapter 5 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

Ch. 5.1 - Prob. 11ECh. 5.1 - Prob. 12ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - Prob. 18ECh. 5.1 - Prob. 19ECh. 5.1 - Prob. 20ECh. 5.1 - Prob. 21ECh. 5.1 - Prob. 22ECh. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Prob. 27ECh. 5.1 - Prob. 28ECh. 5.1 - Prob. 29ECh. 5.1 - Prob. 30ECh. 5.1 - Prob. 31ECh. 5.1 - Prob. 32ECh. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.1 - Prob. 35ECh. 5.1 - Prob. 36ECh. 5.1 - Prob. 37ECh. 5.1 - Prob. 38ECh. 5.1 - Prob. 39ECh. 5.1 - Prob. 40ECh. 5.1 - Prob. 41ECh. 5.1 - Prob. 42ECh. 5.1 - Prob. 43ECh. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 5.1 - Prob. 49ECh. 5.1 - Prob. 50ECh. 5.1 - Prob. 51ECh. 5.1 - Prob. 52ECh. 5.1 - Prob. 53ECh. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Prob. 57ECh. 5.1 - Prob. 58ECh. 5.1 - Prob. 59ECh. 5.1 - Prob. 60ECh. 5.1 - Prob. 61ECh. 5.1 - Prob. 62ECh. 5.1 - Prob. 63ECh. 5.1 - 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Prob. 34ECh. 5.4 - Prob. 35ECh. 5.4 - Prob. 36ECh. 5.4 - Prob. 37ECh. 5.4 - Prob. 38ECh. 5.4 - Prob. 39ECh. 5.4 - Prob. 40ECh. 5.4 - Prob. 41ECh. 5.4 - Prob. 42ECh. 5.4 - Prob. 43ECh. 5.4 - Prob. 44ECh. 5.4 - Prob. 45ECh. 5.4 - Prob. 46ECh. 5.4 - Prob. 47ECh. 5.4 - Prob. 48ECh. 5.4 - Prob. 49ECh. 5.4 - Prob. 50ECh. 5.4 - Prob. 51ECh. 5.4 - Prob. 52ECh. 5.4 - Prob. 53ECh. 5.4 - Prob. 54ECh. 5.4 - Prob. 55ECh. 5.4 - Prob. 56ECh. 5.4 - Prob. 57ECh. 5.4 - Prob. 58ECh. 5.4 - Prob. 59ECh. 5.4 - Prob. 60ECh. 5.4 - Prob. 61ECh. 5.4 - Prob. 62ECh. 5.4 - Prob. 63ECh. 5.4 - Prob. 64ECh. 5.4 - Prob. 65ECh. 5.4 - Prob. 66ECh. 5.4 - Prob. 67ECh. 5.4 - Prob. 68ECh. 5.4 - Prob. 69ECh. 5.4 - Prob. 70ECh. 5.4 - Prob. 71ECh. 5.4 - Prob. 72ECh. 5.4 - Prob. 73ECh. 5.4 - Prob. 74ECh. 5.4 - Prob. 75ECh. 5.4 - Prob. 76ECh. 5.4 - Prob. 77ECh. 5.4 - Prob. 78ECh. 5.4 - Prob. 79ECh. 5.4 - Prob. 80ECh. 5.4 - Prob. 81ECh. 5.4 - Prob. 82ECh. 5.4 - Prob. 83ECh. 5.4 - Prob. 84ECh. 5.4 - Prob. 85ECh. 5.4 - Prob. 86ECh. 5.4 - 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ISBN:9780134763644
Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:PEARSON
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Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman
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Precalculus
Calculus
ISBN:9780135189405
Author:Michael Sullivan
Publisher:PEARSON
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Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning
Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities; Author: Mathispower4u;https://www.youtube.com/watch?v=OmJ5fxyXrfg;License: Standard YouTube License, CC-BY