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 $\theta$ (see figure) can be modeled by $$s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}.$$ (a) Verify that the expression for $s$ is equivalent to $h \cot \theta$ (b) Use a graphing utility to complete the table. Let $h=5$ feet. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 15^{\circ} & 30^{\circ} & 45^{\circ} & 60^{\circ} & 75^{\circ} & 90^{\circ} \\ \hline s & & & & & & \\ \hline \end{array}$$ (c) Use your table from part (b) to determine the angles of the sun that result in the maximum and minimum lengths 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 $\theta$ (see figure) can be modeled by
$$s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}.$$
(a) Verify that the expression for $s$ is equivalent to $h \cot \theta$
(b) Use a graphing utility to complete the table. Let $h=5$ feet.
$$\begin{array}{|l|l|l|l|l|l|l|}
\hline \theta & 15^{\circ} & 30^{\circ} & 45^{\circ} & 60^{\circ} & 75^{\circ} & 90^{\circ} \\
\hline s & & & & & & \\
\hline
\end{array}$$
(c) Use your table from part (b) to determine the angles of the sun that result in the maximum and minimum lengths 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

Cotangent Function

The cotangent of an angle, defined as cot ? = cos ?/sin ?, is an important trigonometric function that naturally arises when comparing the lengths of the adjacent side to the opposite side in a right triangle. This function is particularly useful when modeling ratios and relationships in real-world scenarios like shadow lengths and similar geometric applications.

Interpretation in Real-World Contexts

Mathematical modeling often translates abstract functions into real-world phenomena. Interpreting the results of trigonometric functions, like determining the time of day when the sun reaches particular angles, bridges mathematics and practical observations. This interpretation helps relate optimal or extreme values derived from a model back to real-life experiences or natural phenomena.

Optimization in Trigonometric Functions

Identifying maximum and minimum values of functions is a key aspect of optimization in mathematics. For trigonometric functions, this can involve analyzing function behavior over its domain, recognizing restrictions due to the function's definition, and interpreting the results in a meaningful way, such as understanding the conditions that lead to the shortest or longest shadows.

Trigonometric Identities

Trigonometric identities, such as the complementary angle identity sin(90° - ?) = cos ?, are fundamental tools for rewriting and simplifying expressions in trigonometry. These identities help in transforming one trigonometric function into another, which can simplify the analysis and evaluation of functions in various mathematical problems.

Algebraic Simplification of Trigonometric Expressions

Algebraic manipulation involves using trigonometric identities to transform given expressions into simpler or more recognizable forms. In this context, the process of verifying that a given complex expression is equivalent to a standard form such as h cot ? illustrates the skill of consolidating trigonometric expressions to facilitate easier analysis and computation.

Graphical Analysis and Table Construction

Graphical analysis using tools such as graphing calculators allows one to visually inspect the behavior of functions over specific intervals. Constructing tables based on graphically computed values helps in understanding function trends, analyzing changes based on parameters, and verifying theoretical mathematical results with numerical examples.

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