Height of a Pole A 50 -ft pole casts a shadow as shown in...

Height of a Pole $A 50$ -ft pole casts a shadow as shown in the figure.
(a) Express the angle of elevation $\theta$ of the sun as a function of the length $s$ of the shadow.
(b) Find the angle $\theta$ of elevation of the sun when the shadow is 20 ft long.

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Key Concepts

Angle of Elevation

This is the angle formed between the horizontal line of sight and the line joining the observer's eye to an object, such as the sun. It is core to right triangle problems because it connects the vertical and horizontal measurements in a geometrical setup.

Tangent Function

In trigonometry, the tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to it. This concept is used to relate heights to shadow lengths in problems involving angles of elevation.

Inverse Trigonometric Function

These functions, like arctan (also written as tan?¹), are used to find the angle when given the ratio of two sides of a right triangle. They are crucial in forming an explicit expression for the angle of elevation based on other measurements.

Functional Relationships in Trigonometry

This concept involves expressing one variable as a function of another using trigonometric identities. In geometric problems, particularly those involving shadows and heights, it allows the formulation of equations where the angle of elevation is given as a function of the ratio of the heights or lengths.

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