Solve each problem.See Examples 3 and 4.Angle of Depression of a Light A company safety committee has recommended that a floodlight be mounted in a parking lot so as to illuminate the employee exit, as shown in the figure. Find the angle of depression of the light to the nearest minute.

Verified step by step guidance

1. Identify the right triangle formed by the floodlight, the point on the ground directly below it, and the employee exit.
2. Recognize that the angle of depression from the floodlight to the employee exit is equal to the angle of elevation from the employee exit to the floodlight.
3. Use the tangent function, which relates the opposite side (height of the floodlight) to the adjacent side (horizontal distance to the employee exit), to find the angle of elevation: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
4. Solve for \( \theta \) by taking the arctangent (inverse tangent) of the ratio: \( \theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) \).
5. Convert the angle \( \theta \) from degrees to degrees and minutes, rounding to the nearest minute.

Key Concepts

Angle of Depression

The angle of depression is the angle formed by a horizontal line and the line of sight to an object below that horizontal line. In practical terms, it is measured from the observer's eye level down to the object. Understanding this concept is crucial for solving problems involving heights and distances, particularly in scenarios like the one described, where a floodlight's position relative to the ground is analyzed.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In the context of the angle of depression, these ratios can be used to calculate unknown distances or angles based on known measurements. For example, if the height of the floodlight and the distance to the point of interest are known, trigonometric functions can help determine the angle of depression.

Right Triangle Properties

Right triangle properties are fundamental in trigonometry, as they provide the basis for applying trigonometric ratios. In problems involving angles of depression, the scenario often forms a right triangle where one leg represents the height of the light and the other leg represents the horizontal distance to the point of interest. Recognizing and applying these properties is essential for accurately solving the problem.