Problem 51 - Algebra and Trigonometry

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 of the sun is approximately 38.66 degrees.

Step 1: Understanding the Problem

The height of the tree is provided as 96 ft and the length of the shadow is 120 ft. We are asked to find the angle of elevation of the sun, which is the angle between the line of sight from the top of the tree to the tip of the shadow on the ground and the horizontal ground.

Step 2: Choosing the Right Trigonometric Function

Since we have the opposite side (height of the tree) and the adjacent side (length of the shadow), we can use the tangent function to find the angle. The tangent of an angle in a right triangle is given by the ratio of the opposite side to the adjacent side.

Step 3: Writing the Tangent Formula

Let's denote the angle of elevation as \( \theta \). The formula for tangent is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Here, the opposite side is 96 ft and the adjacent side is 120 ft. So, \( \tan(\theta) = \frac{96}{120} \).

Step 4: Simplifying the Fraction

Simplify the fraction \( \frac{96}{120} \). This can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is 24. Thus, \( \tan(\theta) = \frac{96 \div 24}{120 \div 24} = \frac{4}{5} \).

Step 5: Calculating the Angle

To find the angle \( \theta \), take the arctangent (inverse tangent) of \( \frac{4}{5} \). Use a calculator to find \( \theta = \arctan\left(\frac{4}{5}\right) \). The angle is approximately 38.66 degrees.

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry that relates to the angles and sides of a right triangle. In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. This is mathematically expressed as: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

• Opposite Side: This is the side opposite the angle in question.
• Adjacent Side: The side that forms the angle with the hypotenuse, besides the one you are considering.

The tangent function is particularly useful when solving problems where you are given or need to find one side and an angle in relation to a right triangle. Using the tangent function, you can also solve for the angle if you know the lengths of the opposite and adjacent sides. This was used in our example to determine the angle of elevation of the sun.

Arctangent

Arctangent, often denoted as \( \arctan \) or \( \tan^{-1} \), is the inverse operation of the tangent function. It is used to find the angle whose tangent is a given number. In our example, after establishing that \( \tan(\theta) = \frac{4}{5} \), the arctangent function helps us find the angle \( \theta \).
The arctangent function converts the trigonometric ratio back into an angle, effectively reversing the process of the tangent function. This is useful in scenarios where you have the ratio of the opposite to adjacent sides, like in our tree and shadow problem, and need to find the actual angle involved.
A calculator is often used for this purpose, as it can quickly and accurately compute the angle. In our case, \( \theta \approx 38.66° \) showcases how the arctangent function aids in solving real-world problems involving right triangles.

Right Triangle

Right triangles play a fundamental role in trigonometry and in many practical scenarios like calculating elevations and distances. A right triangle is one that includes a 90-degree angle, which allows for the application of trigonometric functions.

• Hypotenuse: The longest side of a right triangle, opposite the right angle.
• Legs: The other two sides which form the right angle.

In problems involving right triangles, knowing two sides can help you calculate the third side using the Pythagorean theorem, or find an angle using trigonometric ratios such as sine, cosine, or tangent. In our exercise, the right triangle was made between the tree's height, the shadow's length, and the hypotenuse was implied.

Trigonometric Ratio

Trigonometric ratios are tools used to relate angles to the sides of a triangle. They include the sine, cosine, and tangent functions, each defining a unique relationship within right triangles.

• Sine (sin): Ratio of the opposite side to the hypotenuse.
• Cosine (cos): Ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Ratio of the opposite side to the adjacent side.

These ratios allow us to solve for unknown sides or angles in right triangle problems. By isolating one of these functions, you can substitute known values for either angles or sides to determine unknown components. This was key in finding the sun's angle of elevation in our example.