a. Explanation 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 = h sin ( 90 ° − θ ) sin θ . 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 sin θ = cos ( π 2 − θ ) , sin θ = cos ( π 2 − θ ) tan θ = cot ( π 2 − θ ) , cot θ = tan ( π 2 − θ ) sec θ = csc ( π 2 − θ ) , csc θ = sec ( π 2 − θ ) Quotient Identities: tan θ = sin θ cos θ , cot θ = cos θ sin θ Calculation: In order to verify the expression for s , use the cofunction identities and simplify further as shown below, s = h sin ( 90 ° − θ ) sin θ s = h cos θ sin θ ( sin ( 90 ° − θ ) = cos θ ) s = h ⋅ cos θ sin θ s = h ⋅ cot θ s = h cot θ Thus, it verifies that the expression for s is equivalent to h cot θ . b. To determine To complete the table when h = 5 . Answer θ 15 ° 30 ° 45 ° 60 ° 75 ° 90 ° s 18.66 8.66 5 2.89 1.34 0 Explanation 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 = h sin ( 90 ° − θ ) sin θ . Calculation: Using a graphing utility, the value of the s at different angles shown below, θ 15 ° 30 ° 45 ° 60 ° 75 ° 90 ° s 18.66 8.66 5 2.89 1.34 0 c. To determine To verify that the expression for s is equivalent to h cot θ . Answer Maximum length at: 15 ° Minimum length at: 90 ° Explanation 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 = h sin ( 90 ° − θ ) sin θ . 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 ° . Answer Noon 12: 00 PM Explanation 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 = h sin ( 90 ° − θ ) sin θ . 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.

Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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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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Question

Chapter 5.2, Problem 79E

To determine

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

Expert Solution

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 $hcotθ$ .

To determine

To complete the table when $h=5$ .

Expert Solution

Answer to Problem 79E

$θ$	$15°$	$30°$	$45°$	$60°$	$75°$	$90°$
s	18.66	8.66	5	2.89	1.34	0

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,

$θ$	$15°$	$30°$	$45°$	$60°$	$75°$	$90°$
s	18.66	8.66	5	2.89	1.34	0

To determine

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

Expert Solution

Answer to Problem 79E

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

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 $90°$ .

To determine

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

Expert Solution

Answer to Problem 79E

$Noon 12:00 PM$

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 5.2, Problem 78E

Chapter 5.2, Problem 80E

Chapter 5 Solutions

Precalculus with Limits: A Graphing Approach

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Prob. 70E Ch. 5.5 - Prob. 71E Ch. 5.5 - Prob. 72E Ch. 5.5 - Prob. 73E Ch. 5.5 - Prob. 74E Ch. 5.5 - Prob. 75E Ch. 5.5 - Prob. 76E Ch. 5.5 - Prob. 77E Ch. 5.5 - Prob. 78E Ch. 5.5 - Prob. 79E Ch. 5.5 - Prob. 80E Ch. 5.5 - Prob. 81E Ch. 5.5 - Prob. 82E Ch. 5.5 - Prob. 83E Ch. 5.5 - Prob. 84E Ch. 5.5 - Prob. 85E Ch. 5.5 - Prob. 86E Ch. 5.5 - Prob. 87E Ch. 5.5 - Prob. 88E Ch. 5.5 - Prob. 89E Ch. 5.5 - Prob. 90E Ch. 5.5 - Prob. 91E Ch. 5.5 - Prob. 92E Ch. 5.5 - Prob. 93E Ch. 5.5 - Prob. 94E Ch. 5.5 - Prob. 95E Ch. 5.5 - Prob. 96E Ch. 5.5 - Prob. 97E Ch. 5.5 - Prob. 98E Ch. 5.5 - Prob. 99E Ch. 5.5 - Prob. 100E Ch. 5.5 - Prob. 101E Ch. 5.5 - Prob. 102E Ch. 5.5 - Prob. 103E Ch. 5.5 - Prob. 104E Ch. 5.5 - Prob. 105E Ch. 5.5 - Prob. 106E Ch. 5.5 - Prob. 107E Ch. 5.5 - Prob. 108E Ch. 5.5 - Prob. 109E Ch. 5.5 - Prob. 110E Ch. 5.5 - Prob. 111E Ch. 5.5 - Prob. 112E Ch. 5.5 - Prob. 113E Ch. 5.5 - Prob. 114E Ch. 5.5 - Prob. 115E Ch. 5.5 - Prob. 116E Ch. 5.5 - Prob. 117E Ch. 5.5 - Prob. 118E Ch. 5.5 - Prob. 119E Ch. 5.5 - Prob. 120E Ch. 5.5 - Prob. 121E Ch. 5.5 - Prob. 122E Ch. 5.5 - Prob. 123E Ch. 5.5 - Prob. 124E Ch. 5.5 - Prob. 125E Ch. 5.5 - Prob. 126E Ch. 5.5 - Prob. 127E Ch. 5.5 - Prob. 128E Ch. 5.5 - Prob. 129E Ch. 5.5 - Prob. 130E Ch. 5.5 - Prob. 131E Ch. 5.5 - Prob. 132E Ch. 5.5 - Prob. 133E Ch. 5.5 - Prob. 134E Ch. 5.5 - Prob. 135E Ch. 5.5 - Prob. 136E Ch. 5.5 - Prob. 137E Ch. 5.5 - Prob. 138E Ch. 5.5 - Prob. 139E Ch. 5.5 - Prob. 140E Ch. 5 - Prob. 1RE Ch. 5 - Prob. 2RE Ch. 5 - Prob. 3RE Ch. 5 - Prob. 4RE Ch. 5 - Prob. 5RE Ch. 5 - Prob. 6RE Ch. 5 - Prob. 7RE Ch. 5 - Prob. 8RE Ch. 5 - Prob. 9RE Ch. 5 - Prob. 10RE Ch. 5 - Prob. 11RE Ch. 5 - Prob. 12RE Ch. 5 - Prob. 13RE Ch. 5 - Prob. 14RE Ch. 5 - Prob. 15RE Ch. 5 - Prob. 16RE Ch. 5 - Prob. 17RE Ch. 5 - Prob. 18RE Ch. 5 - Prob. 19RE Ch. 5 - Prob. 20RE Ch. 5 - Prob. 21RE Ch. 5 - Prob. 22RE Ch. 5 - Prob. 23RE Ch. 5 - Prob. 24RE Ch. 5 - Prob. 25RE Ch. 5 - Prob. 26RE Ch. 5 - Prob. 27RE Ch. 5 - Prob. 28RE Ch. 5 - Prob. 29RE Ch. 5 - Prob. 30RE Ch. 5 - Prob. 31RE Ch. 5 - Prob. 32RE Ch. 5 - Prob. 33RE Ch. 5 - Prob. 34RE Ch. 5 - Prob. 35RE Ch. 5 - Prob. 36RE Ch. 5 - Prob. 37RE Ch. 5 - Prob. 38RE Ch. 5 - Prob. 39RE Ch. 5 - Prob. 40RE Ch. 5 - Prob. 41RE Ch. 5 - Prob. 42RE Ch. 5 - Prob. 43RE Ch. 5 - Prob. 44RE Ch. 5 - Prob. 45RE Ch. 5 - Prob. 46RE Ch. 5 - Prob. 47RE Ch. 5 - Prob. 48RE Ch. 5 - Prob. 49RE Ch. 5 - Prob. 50RE Ch. 5 - Prob. 51RE Ch. 5 - Prob. 52RE Ch. 5 - Prob. 53RE Ch. 5 - Prob. 54RE Ch. 5 - Prob. 55RE Ch. 5 - Prob. 56RE Ch. 5 - Prob. 57RE Ch. 5 - Prob. 58RE Ch. 5 - Prob. 59RE Ch. 5 - Prob. 60RE Ch. 5 - Prob. 61RE Ch. 5 - Prob. 62RE Ch. 5 - Prob. 63RE Ch. 5 - Prob. 64RE Ch. 5 - Prob. 65RE Ch. 5 - Prob. 66RE Ch. 5 - Prob. 67RE Ch. 5 - Prob. 68RE Ch. 5 - Prob. 69RE Ch. 5 - Prob. 70RE Ch. 5 - Prob. 71RE Ch. 5 - Prob. 72RE Ch. 5 - Prob. 73RE Ch. 5 - Prob. 74RE Ch. 5 - Prob. 75RE Ch. 5 - Prob. 76RE Ch. 5 - Prob. 77RE Ch. 5 - Prob. 78RE Ch. 5 - Prob. 79RE Ch. 5 - Prob. 80RE Ch. 5 - Prob. 81RE Ch. 5 - Prob. 82RE Ch. 5 - Prob. 83RE Ch. 5 - Prob. 84RE Ch. 5 - Prob. 85RE Ch. 5 - Prob. 86RE Ch. 5 - Prob. 87RE Ch. 5 - Prob. 88RE Ch. 5 - Prob. 89RE Ch. 5 - Prob. 90RE Ch. 5 - Prob. 91RE Ch. 5 - Prob. 92RE Ch. 5 - Prob. 93RE Ch. 5 - Prob. 94RE Ch. 5 - Prob. 95RE Ch. 5 - Prob. 96RE Ch. 5 - Prob. 97RE Ch. 5 - Prob. 98RE Ch. 5 - Prob. 99RE Ch. 5 - Prob. 100RE Ch. 5 - Prob. 101RE Ch. 5 - Prob. 102RE Ch. 5 - Prob. 103RE Ch. 5 - Prob. 104RE Ch. 5 - Prob. 105RE Ch. 5 - Prob. 106RE Ch. 5 - Prob. 107RE Ch. 5 - Prob. 108RE Ch. 5 - Prob. 109RE Ch. 5 - Prob. 110RE Ch. 5 - Prob. 111RE Ch. 5 - Prob. 112RE Ch. 5 - Prob. 113RE Ch. 5 - Prob. 114RE Ch. 5 - Prob. 115RE Ch. 5 - Prob. 116RE Ch. 5 - Prob. 117RE Ch. 5 - Prob. 118RE Ch. 5 - Prob. 119RE Ch. 5 - Prob. 120RE Ch. 5 - Prob. 121RE Ch. 5 - Prob. 122RE Ch. 5 - Prob. 123RE Ch. 5 - Prob. 124RE Ch. 5 - Prob. 125RE Ch. 5 - Prob. 126RE Ch. 5 - Prob. 127RE Ch. 5 - Prob. 128RE Ch. 5 - Prob. 129RE Ch. 5 - Prob. 130RE Ch. 5 - Prob. 131RE Ch. 5 - Prob. 132RE Ch. 5 - Prob. 133RE Ch. 5 - Prob. 134RE Ch. 5 - Prob. 135RE Ch. 5 - Prob. 136RE Ch. 5 - Prob. 137RE Ch. 5 - Prob. 138RE Ch. 5 - Prob. 139RE Ch. 5 - Prob. 140RE Ch. 5 - Prob. 1CT Ch. 5 - Prob. 2CT Ch. 5 - Prob. 3CT Ch. 5 - Prob. 4CT Ch. 5 - Prob. 5CT Ch. 5 - Prob. 6CT Ch. 5 - Prob. 7CT Ch. 5 - Prob. 8CT Ch. 5 - Prob. 9CT Ch. 5 - Prob. 10CT Ch. 5 - Prob. 11CT Ch. 5 - Prob. 12CT Ch. 5 - Prob. 13CT Ch. 5 - Prob. 14CT Ch. 5 - Prob. 15CT Ch. 5 - Prob. 16CT Ch. 5 - Prob. 17CT Ch. 5 - Prob. 18CT Ch. 5 - Prob. 19CT Ch. 5 - Prob. 20CT Ch. 5 - Prob. 21CT Ch. 5 - Prob. 22CT Ch. 5 - Prob. 23CT Ch. 5 - Prob. 24CT

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