The length s of a shadow cast by a vertical gnomon (a device...

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 $s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}$ $0^{\circ}<\theta \leq 90^{\circ}.$ (a) Verify that the expression for $s$ is equal to $h \cot \theta$ (b) Use a graphing utility to create a table of the lengths $s$ for different values of $\theta .$ Let $h=5$ feet. (c) Use your table from part (b) to determine the angle of the sun that results in the minimum length of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is $90^{\circ} ?$

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 $s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}$ $0^{\circ}<\theta \leq 90^{\circ}.$
(a) Verify that the expression for $s$ is equal to $h \cot \theta$
(b) Use a graphing utility to create a table of the lengths $s$ for different values of $\theta .$ Let $h=5$ feet.
(c) Use your table from part (b) to determine the angle of the sun that results in the minimum length of the shadow.
(d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is $90^{\circ} ?$

Key Concepts

Trigonometric Identities

This concept involves understanding and applying the basic relationships between trigonometric functions. In many problems, identities like sin(90° - ?) = cos ? are used to simplify expressions, making it easier to solve equations or transform functions into more useful forms.

Trigonometric Ratios and Functions

This includes a thorough grasp of functions such as sine, cosine, tangent, and their reciprocals like cotangent. Recognizing that cot ? is defined as cos ?/sin ? is key to both simplifying expressions and interpreting various physical models that relate angles to side lengths.

Graphical Analysis of Functions

Graphical analysis involves using visual tools, like graphing utilities and data tables, to explore how functions behave over specific intervals. This technique is essential for identifying trends, verifying analytical results, and determining extremum values such as minimum or maximum points within a model.

Optimization in Trigonometric Models

Optimization refers to the process of finding the best value (such as a minimum or maximum) of a function under given conditions. In the context of trigonometric models, it involves analyzing how changes in the angle affect outcomes, which is a key step in solving problems that require determining optimal conditions.

Transcript

00:01 For this question, we are given our recreation s, which is the length of the shadow, and it's equal to h times sine of 90 degrees minus theta over sine theta.

00:23 Okay, so now part a of the question, we're asked to verify that the expression for s is equal to h through tangent theta.

00:33 So to do this, we need to use our co -function, which actually tells us sine of 90 degrees minus theta is equal to cosine theta.

00:48 Okay, so now with that, we can see that s is equal to h times cos theta over sine theta.

00:59 And cos theta over sine theta is just cotangent theta.

01:02 So this is h times co -hanging data.

01:07 Okay, and we're done.

01:09 Now, part b, we're asked to use a graphing utility to create a table of length s for different values of theta.

01:18 And we're going to use h as equal to 5 feet.

01:23 Okay, so let's go over to our graphing utility.

01:29 Okay, so i know here this data, but it's just easier if we use in part x.

01:34 So let's say s is a function of x, so we're just replacing theta with x.

01:39 And this is equal to h, which is 5 in this case, times co -tangent of theta, which is just x.

01:47 So this is what your graph should look like.

01:49 If it looks a bit different, make sure that your calculator is set to degrees, because if you have it in radiance, it will look something like this.

01:58 So we want to make sure it's in degrees because the question is given in degrees...

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