Problem 162 - Precalculus

Question

A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

Step-by-Step Solution

Answer

The acute angle adjacent to the 4-foot side is approximately 18.43°.

Step 1: Understand the Triangle Problem

We need to find one of the acute angles of a right triangle. The right triangle has a base of 12 feet and a height of 4 feet. The angles of interest are adjacent to the height, which is 4 feet.

Step 2: Recognize the Sides of the Triangle

In this right triangle, the base (12 feet) is adjacent to the angle we want to find. Thus, the known sides here are opposite (4 feet) and adjacent (12 feet) to our angle of interest.

Step 3: Use Trigonometric Ratios to Find the Angle

To find this angle, we use the tangent function, which relates the opposite side to the adjacent side. The formula is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Substitute the given side lengths: \( \tan(\theta) = \frac{4}{12} = \frac{1}{3} \).

Step 4: Calculate the Angle Using Inverse Tangent

To find the angle itself, use the inverse tangent function. \( \theta = \tan^{-1}\left(\frac{1}{3}\right) \). Use a calculator to find \( \theta \approx 18.43^\circ \).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The other two angles are acute, meaning they are less than 90 degrees. These triangles are essential in trigonometry because they allow us to establish relationships between the angles and sides.
A right triangle consists of three sides:

• The hypotenuse, which is the longest side and is opposite the right angle.
• The base, commonly referred to as one of the legs of the triangle, adjacent to the right angle.
• The height, also a leg, perpendicular to the base and adjacent to the right angle.

In our exercise, we focus on one of these right triangles with a base of 12 feet and a height of 4 feet. Each side plays a crucial role when determining angles using trigonometric functions. If you're given side lengths, you can always find the missing angles with the help of these principles.

Tangent Function

The tangent function is one of the basic trigonometric functions, alongside sine and cosine. It is particularly useful in right triangles as it helps us find and relate the unknown angles to known side lengths.
The formula for the tangent of an angle \( \theta \) in a right triangle is:

• \( \tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}} \)

In the current problem, we want to find the angle adjacent to the height of 4 feet. With the opposite side being 4 feet and the adjacent side being 12 feet, our tangent function becomes \( \tan(\theta) = \frac{4}{12} = \frac{1}{3} \).
This function is useful because it provides a direct way of calculating the angle when the ratio of the side lengths is given, which forms the basis for using inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as inverse tangent (often denoted as \( \tan^{-1} \) or arctan), are essential when we want to calculate angles in right triangles when we know the side ratios.
The inverse tangent function allows us to "reverse" the tangent process, effectively finding the unknown angle when we already know the tangent value. For our problem, we found \( \tan(\theta) = \frac{1}{3} \), so to find \( \theta \):

• Use \( \theta = \tan^{-1}\left(\frac{1}{3}\right) \)

Using a calculator, this yields an angle of approximately 18.43 degrees. This value represents one of the acute angles in our right triangle, demonstrating the power and utility of inverse trigonometric functions in practical applications.