Solve each problem.See Examples 3 and 4. Length of a Shadow Suppose that the angle of elevation of the sun is 23.4°. Find the length of the shadow cast by a person who is 5.75 ft tall.​​

Verified step by step guidance

1. Step 1: Understand the problem setup. The angle of elevation of the sun is given as 23.4°, and the height of the person is 5.75 ft. We need to find the length of the shadow cast by the person.
2. Step 2: Visualize the scenario as a right triangle. The person's height is the opposite side, the shadow is the adjacent side, and the angle of elevation is the angle between the shadow and the line of sight to the sun.
3. Step 3: Use the tangent function, which relates the opposite side and the adjacent side in a right triangle. The formula is $\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}$.
4. Step 4: Substitute the known values into the tangent formula. Here, $\theta =23.4°$ and the opposite side (height of the person) is 5.75 ft. So, $\tan (23.4°)=\frac{5.75}{\text{shadow length}}$.
5. Step 5: Solve for the shadow length by rearranging the equation: $\text{shadow length}=\frac{5.75}{\tan (23.4°)}$.

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed by the horizontal line and the line of sight to an object above that line. In this context, it refers to the angle at which the sun's rays strike the ground, affecting the length of the shadow cast by an object. Understanding this angle is crucial for applying trigonometric functions to solve problems involving heights and distances.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a right triangle to the lengths of its sides. In this problem, the tangent function is particularly relevant, as it relates the height of the person to the length of the shadow. Specifically, the tangent of the angle of elevation equals the opposite side (height) over the adjacent side (shadow length).

Right Triangle Properties

A right triangle has one angle measuring 90 degrees, and the relationships between its sides and angles are governed by trigonometric principles. In this scenario, the person, their shadow, and the line from the top of their head to the tip of the shadow form a right triangle. Recognizing this structure allows for the application of trigonometric functions to find unknown lengths, such as the shadow's length.