Problem 38 - Precalculus Mathematics for Calculus

Question

A 96-ft tree casts a shadow that is 120 ft long. What is the angle of elevation of the sun?

Step-by-Step Solution

Answer

The angle of elevation is approximately 38.66 degrees.

Step 1: Understand the problem

We need to find the angle of elevation of the sun when a 96-ft tall tree casts a shadow that is 120 ft long. This involves using trigonometry with the tangent function.

Step 2: Identify the right triangle components

A right triangle can be formed with the tree as the opposite side, the shadow as the adjacent side, and the angle of elevation as the angle between the ground and the line from the shadow's tip to the top of the tree.

Step 3: Apply trigonometric ratio for tangent

The tangent of an angle in a right triangle is given by the ratio of the opposite side to the adjacent side. Here, \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{96}{120} \]

Step 4: Simplify the tangent ratio

Simplify the ratio to make calculations easier: \[ \tan(\theta) = \frac{96}{120} = \frac{4}{5} \]

Step 5: Calculate the angle of elevation

Use the arctangent (inverse tangent) function to calculate the angle: \[ \theta = \arctan\left(\frac{4}{5}\right) \] This requires using a calculator set to degrees.

Step 6: Convert the arctangent result

Calculate \( \theta \) using a calculator: \[ \theta \approx 38.66^{\circ} \] Therefore, the angle of elevation of the sun is approximately 38.66 degrees.

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry used to relate angles to the lengths of sides in a right triangle. When dealing with tangent, you have the three key components: the angle, the opposite side, and the adjacent side. The tangent of an angle \( \theta \) is calculated as the ratio of the length of the opposite side to the length of the adjacent side:

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

In the context of elevation problems, the opposite side is usually the height of an object, while the adjacent side is the length of the shadow it casts. Knowing this ratio helps solve for unknown dimensions in right triangles. For example, in our tree problem, we use the tangent ratio to find the angle of elevation of the sun based on the tree height and the shadow length.

Right Triangle

A right triangle is a triangle where one of its angles is exactly 90 degrees. This special type of triangle is crucial in trigonometry because it allows us to understand the relationships between angles and side lengths. In trigonometric calculations:

• The side opposite to the right angle is known as the hypotenuse, which is always the longest side.
• The other two sides, which we refer to when calculating trig functions like tangent, are the opposite and adjacent sides.

In our example, the tree, the shadow, and the line from the top of the tree to the tip of the shadow form a right triangle. Understanding this helps identify which trigonometric functions to apply and is foundational for calculating unknown angles or lengths.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. It provides tools to solve problems involving heights and distances, among others. The three primary functions in trigonometry are sine, cosine, and tangent:

• Sine: Ratio of the opposite side to the hypotenuse.
• Cosine: Ratio of the adjacent side to the hypotenuse.
• Tangent: Ratio of the opposite side to the adjacent side.

Each of these functions provides a way to find unknown angles when we know some side lengths, or vice versa. In the angle of elevation problem, trigonometry enables us to use the tangent function to calculate the angle based on the available side lengths—demonstrating not only the core idea of trigonometry but also its practical applications in real-world scenarios.