Problem 35 - Precalculus: Functions and Graphs

Question

Approximate the angle of elevation \(\alpha\) of the sun if a person 5.0 feet tall casts a shadow 4.0 feet long on level ground (see the figure). (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The angle of elevation \( \alpha \) is approximately \( 51.34^{\circ} \).

Step 1: Understanding the Problem

We need to approximate the angle of elevation \( \alpha \) of the sun. A person who is 5.0 feet tall casts a shadow that is 4.0 feet long. This problem can be solved using trigonometry and the concept of similar triangles found in the right triangle formed by the person, their shadow, and the line of sight to the sun.

Step 2: Identifying Trigonometric Relation

In a right triangle, the tangent of the angle of elevation is the ratio of the opposite side (height of the person) to the adjacent side (length of the shadow). This can be expressed as: \( \tan(\alpha) = \frac{\text{height of person}}{\text{length of shadow}} = \frac{5.0}{4.0} \).

Step 3: Calculating the Tangent Ratio

Calculate the ratio: \( \tan(\alpha) = \frac{5.0}{4.0} = 1.25 \). This ratio represents the tangent of the angle of elevation.

Step 4: Finding the Angle of Elevation

To find \( \alpha \), use the inverse tangent function, which is also called the arctan function: \( \alpha = \tan^{-1}(1.25) \).

Step 5: Calculating the Angle

Use a calculator to compute \( \alpha = \tan^{-1}(1.25) \approx 51.34^{\circ} \). This is the angle of elevation of the sun.

Key Concepts

Angle of Elevation

When you hear the term "angle of elevation," think about looking up towards the sky. Imagine you're standing and looking at the sun. The angle of elevation is the angle between your line of sight and the horizontal ground. This concept becomes handy when you want to measure how high something is above the ground, like a flagpole or, as in this case, the sun when you know how long the shadow is. Whenever you measure upwards from a horizontal level, that's where the angle of elevation comes into play.

In this specific exercise, understanding the angle of elevation helps us figure out how high above the horizon the sun appears when it casts your shadow on the ground. The larger the angle, the higher the sun is in the sky.

Right Triangle

A right triangle is a triangle where one of the angles is exactly 90 degrees. It's a crucial part of this exercise. When a person stands on the ground and the sun casts their shadow, a right triangle forms:

• The person's height forms the vertical side of the triangle.
• The shadow acts as the horizontal base.
• The line of sight to the sun acts as the hypotenuse.

This triangle allows us to apply trigonometric rules. Right triangles are useful because they make it easier to use certain trigonometric functions to calculate unknown values, such as angles or side lengths. Remember, any time you see a right angle, you can find valuable trigonometric relationships.

Tangent Function

The tangent function is one of the primary trigonometric functions. In right triangles, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For this exercise:

• The person's height is the opposite side.
• The length of the shadow is the adjacent side.

Thus, we express it as: \[ an( heta) = \frac{\text{Opposite}}{\text{Adjacent}}. \] Substituting the values from the exercise, we have \[ \tan(\alpha) = \frac{5.0}{4.0} = 1.25. \]This ratio tells you how steep your climb is if you're looking from the shadow's tip to the top of the person. A larger tangent value indicates a steeper angle, meaning the sun is higher in the sky.

Inverse Trigonometric Functions

Once you have the tangent value, the next step is to find the angle itself, which is where inverse trigonometric functions come in. For angle calculations, you specifically use the inverse tangent function, commonly referred to as arctan or \(\tan^{-1}\). It helps reverse the tangent operation to solve for the angle:

• If \( \tan(\alpha) = 1.25 \), then \( \alpha = \tan^{-1}(1.25) \).

This function gives you the angle whose tangent is 1.25.

Inverse trigonometric functions like arctan are valuable because they allow you to find angles when you know the ratio of the sides in a right triangle. In this exercise, using a calculator, you discover that \( \alpha \) is approximately 51.34 degrees. This result represents the sun's elevation angle above the horizontal when casting your 4-foot shadow.