66. Shadow Length. The length s of a shadow cast by a...

66. Shadow Length. The length $s$ of a shadow cast by a vertical gnomon (a device used to tell time) of height $h$ when the angle of the sun above the horizon is $\theta$ can be modeled by the equation $$s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}.$$ $\begin{array}{l}{\text { (a) Verify that the expression for } s \text { is equal to } h \text { cot } \theta \text { . }} \\ {\text { (b) Use a graphing utility to complete the table. Let }} {h=5 \text { feet. }}\end{array}$ $\begin{array}{l}{\text { (c) Use your table from part (b) to determine the }} \\ {\text { angles of the sun that result in the maximum and }} \\ {\text { minimum lengths of the shadow. }} \\ {\text { (d) Based on your results from part (c), what time of }} \\ {\text { day do you think it is when the angle of the sun }} \\ {\text { above the horizon is } 90^{\circ} ?}\end{array}$

66. Shadow Length. The length $s$ of a shadow cast by a vertical gnomon (a device used to tell time) of height $h$ when the angle of the sun above the horizon is $\theta$ can be modeled by the equation

$$s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}.$$

$\begin{array}{l}{\text { (a) Verify that the expression for } s \text { is equal to } h \text { cot } \theta \text { . }} \\ {\text { (b) Use a graphing utility to complete the table. Let }} {h=5 \text { feet. }}\end{array}$

$\begin{array}{l}{\text { (c) Use your table from part (b) to determine the }} \\ {\text { angles of the sun that result in the maximum and }} \\ {\text { minimum lengths of the shadow. }} \\ {\text { (d) Based on your results from part (c), what time of }} \\ {\text { day do you think it is when the angle of the sun }} \\ {\text { above the horizon is } 90^{\circ} ?}\end{array}$

Key Concepts

Trigonometric Identities

Trigonometric identities, such as the complementary angle identity sin(90° - ?) = cos ?, provide the means to simplify and manipulate trigonometric expressions. In this context, using the identity transforms the given expression for the shadow length into a form that involves the cotangent function, enabling further analysis and interpretation.

Cotangent Function

The cotangent function is defined as the ratio of the cosine of an angle to the sine of that angle (cot ? = cos ?/sin ?). This concept is used to express the shadow length in a simplified form and is essential for understanding how the length of the shadow changes with varying sun angles.

Graphical Analysis of Functions

Graphing functions, especially trigonometric functions, helps determine the behavior of the function over a particular domain, including the identification of maximum and minimum values. Using a graphing utility to plot the shadow length as a function of the sun's angle is an example of applying this concept to investigate how the shadow length varies throughout the day.

Mathematical Modeling in Physics

Mathematical modeling involves representing physical scenarios, such as the casting of shadows, using mathematical functions. Understanding how trigonometric functions relate physical quantities like the height of a gnomon and the angle of the sun to the length of the shadow is key to predicting and analyzing real-world phenomena.

Transcript

00:04 So here we have a formula denotes the length of a shadow produced right by a vertical nomon.

00:20 So a vertical no -mon think of it as like a sun dial.

00:24 So h refers to the height of the vertical no -moon, right? and then s would refer to the length, right, of the shadow produced.

00:34 All right, so the sun hits the nomaddle.

00:38 All right.

00:39 Okay, so the first thing that we need to consider is, okay, can we reduce this to a simpler trigonometric function? i think we can.

00:51 So let's try to simplify this.

00:53 Let's start off with sine of 90 minus data.

00:57 So this one is a co -function identity, right, which states that a trigonometric function for a particular angle.

01:11 Is as the same value, all right, as the co -function value to the complement of the angle.

01:21 All right.

01:21 Now, for sign, right, the co -function of sine is actually cosine.

01:30 So therefore, sine of 90 minus theta would be equivalent to cosine, right? because cosine is the co -function of sign all right of the complement of the angle all right so therefore if one angle is 90 minus theta right but you need to add to this angle to make it 90 degrees is actually just going to be theta all right so therefore s can be expressed as h cosine of theta over sine data.

02:09 All right.

02:10 Now we have here the ratio cosine over sign data.

02:17 This ratio of course can be expressed as simply cotangent of data and this would be of course a simpler trigonometric function to relate the shadow formed right based from the angle the sun is located above the horizon and the height of the mom.

02:45 All right.

02:46 Okay.

02:47 So next, right, let's try to like plot some values for this function and see, you know, what the lengths of the shadows are at particular angle measures.

03:06 All right.

03:08 Okay.

03:09 So here we go...

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