Problem 32 - Algebra and Trigonometry

Question

A security camera in a neighborhood bank is mounted on a wall 9 feet above the floor. What angle of depression should be used if the camera is to be directed to a spot 6 feet above the floor and 12 feet from the wall?

Step-by-Step Solution

Answer

The angle of depression should be approximately 14.04°.

Step 1: Identify the right triangle

Visualize and draw a right triangle formed by the height from the wall to the spot, the horizontal distance from the wall, and the hypotenuse (line of sight from the camera to the spot). The vertical side of the triangle is the difference in height, which is 9 feet - 6 feet = 3 feet. The horizontal side is given as 12 feet.

Step 2: Set up the trigonometric function

Use the tangent function for the angle of depression, which relates the opposite side (vertical distance) to the adjacent side (horizontal distance). The formula is \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{12} \].

Step 3: Solve for the angle of depression

Calculate the arctangent (inverse tangent) of the ratio \( \frac{3}{12} \) using \[ \theta = \tan^{-1}(\frac{3}{12}) \]. Using a calculator, \[ \theta \approx \tan^{-1}(0.25) \approx 14.04^\circ \].

Key Concepts

angle of depression

The angle of depression is a key concept in this problem. It refers to the angle formed by the line of sight when an observer looks downward from a horizontal line. Imagine standing on a high point and looking down to the ground; the angle between your horizontal line of sight and the line of sight to the ground is the angle of depression.

To find the angle of depression, you can use various trigonometric functions. In this exercise, we specifically use the tangent function, since we have a right triangle.

right triangle

A right triangle is a triangle that has one angle equal to 90 degrees. This type of triangle is fundamental in trigonometry because it allows for the use of trigonometric functions to relate the angles to their side lengths.

In the given exercise, we visualize a right triangle with:

• The vertical side (difference in height) being 3 feet (9 feet - 6 feet).
• The horizontal side (distance from the wall) being 12 feet.
• The hypotenuse (line of sight from the camera to the spot).

With these measurements, we can apply the tangent function to find the angle of depression.

tangent function

The tangent function, denoted as \tan(\theta), is a trigonometric function that relates the angles of a right triangle to the ratios of two of its sides. Specifically, for any angle \theta in a right triangle,

\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Here, 'opposite' refers to the side opposite the angle in question, and 'adjacent' refers to the side next to the angle.

For our exercise, the angle \theta is the angle of depression, the vertical side (opposite) is 3 feet, and the horizontal side (adjacent) is 12 feet. Thus, we set it up as:\( \tan(\theta) = \frac{3}{12} \approx 0.25 \).

Understanding this relationship makes it simpler to now solve for \theta using the arctangent function.

arctangent

The arctangent function (also known as the inverse tangent function) is used to determine an angle when the tangent of that angle is known. It is denoted as \tan^{-1} or \text{arctan}.

To find the angle \theta in the exercise, we use:
\( \theta = \tan^{-1} \bigg(\frac{3}{12}\bigg) \)
Understanding arctangent helps us reverse-engineer the tangent function, taking the ratio of the opposite to adjacent sides to find the actual angle. Utilizing a calculator, we get:
\( \theta \approx \tan^{-1}(0.25) \approx 14.04^\text{o} \).

Thus, the angle of depression needed for the camera to be directed to the spot is approximately 14.04 degrees.