Problem 38 - Algebra and Trigonometry

Question

Angle of the Sun 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 of the sun is approximately 38.66 degrees.

Step 1: Understand the Problem

Visualize the scenario as a right triangle where the tree forms the opposite side, the shadow forms the adjacent side, and the angle of elevation is the angle formed by the ground and the line of sight from the top of the tree to the tip of the shadow.

Step 2: Identify the Trigonometric Function

To find the angle of elevation, we use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Here, \(\theta\) is the angle of elevation.

Step 3: Substitute Known Values

Plug in the known values into the tangent formula. For this problem, the opposite side (tree height) is 96 ft, and the adjacent side (shadow length) is 120 ft. So, \( \tan(\theta) = \frac{96}{120} \).

Step 4: Simplify the Tangent Ratio

Simplify the fraction \( \frac{96}{120} \) by dividing the numerator and the denominator by their greatest common divisor, which is 24: \( \frac{96}{120} = \frac{4}{5} \).

Step 5: Calculate the Angle of Elevation

Use the arctangent function to find the angle: \( \theta = \tan^{-1}\left(\frac{4}{5}\right) \). Calculate this using a calculator to find \( \theta \approx 38.66\degree \).

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between the horizontal ground and the line of sight when looking at an object above the ground. For instance, when the sun shines on a tall tree, and a shadow is cast, we can contemplate this scenario as a right triangle. Here, the angle from the tip of the shadow to the top of the tree represents the angle of elevation.
This concept is important in various real-world applications like architecture, navigation, and astronomy. Calculating it helps determine heights of objects or distances that cannot be measured directly.

Tangent Function

The tangent function is a trigonometric function that relates the angle in a right triangle to the lengths of the opposite side and the adjacent side. Mathematically, it is expressed as:

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

In the context of the exercise, the tree and its shadow form a right triangle with the ground, where the height of the tree is the opposite side and the length of the shadow is the adjacent side. The tangent function allows us to find the angle of elevation by using these two measurements.

Right Triangle

A right triangle is a type of triangle where one angle is precisely 90 degrees. This property makes right triangles particularly useful for applying trigonometric functions, like sine, cosine, and tangent, to solve problems in geometry and physics.
In this exercise, if you visualize the scenario, the right triangle consists of:

• The tree as the opposite side.
• The shadow as the adjacent side.
• The ground forming the right angle.

Understanding this setup allows us to effectively use the tangent function for finding other unknown elements, like angles.

Arctangent

The arctangent, denoted as \( \tan^{-1} \), is the inverse function of the tangent. It is used to calculate the angle when the ratio of the opposite to adjacent sides is known. This makes it a handy tool in trigonometry to find angles in a right triangle.
For our exercise, once the tangent of the angle of elevation is identified as \( \frac{4}{5} \) from the tree's height and shadow's length, we use the arctangent function to find \( \theta \), the angle of elevation:

• \( \theta = \tan^{-1}\left(\frac{4}{5}\right) \)

This process converts the tangent ratio back into an angle measurement, yielding an answer like \( \theta \approx 38.66\degree \), providing the desired angle.