Problem 37 - A Graphical Approach to Precalculus with Limits

Question

A tunnel is to be dug from \(A\) to \(B\). Both \(A\) and \(B\) are visible from \(C\). If \(A C\) is 1.4923 miles, \(B C\) is 1.0837 miles, and \(C\) is exactly \(90^{\circ}\) find the measures of angles \(A\) and \(B\)

Step-by-Step Solution

Answer

Angle A is approximately 35.880° and Angle B is approximately 54.120°.

Step 1: Understand the Triangle Configuration

Visualize the problem as a right triangle \( \triangle ABC \) where \( C \) is the right angle. The sides are \( AC = 1.4923 \) miles and \( BC = 1.0837 \) miles. The task is to find angles \( \angle A \) and \( \angle B \).

Step 2: Use Trigonometry for Angle A

To find \( \angle A \), use the tangent function: \( \tan A = \frac{opposite}{adjacent} \). Here, \( opposite = BC = 1.0837 \) miles and \( adjacent = AC = 1.4923 \) miles. Thus, \( \tan A = \frac{1.0837}{1.4923} \).

Step 3: Calculate Angle A

Solve for \( \angle A \) by calculating \( A = \tan^{-1} \left( \frac{1.0837}{1.4923} \right) \). Use a calculator to find the angle in degrees: \( \angle A \approx 35.880 \) degrees.

Step 4: Find Angle B Using Angle Sum Property

Since the sum of angles in a triangle is \( 180^{\circ} \), and angle \( C = 90^{\circ} \), angle \( B \) can be found using \( B = 90^{\circ} - A \).

Step 5: Calculate Angle B

Compute \( \angle B = 90^{\circ} - 35.880^{\circ} \approx 54.120^{\circ} \). This gives us the measure of \( \angle B \).

Key Concepts

Trigonometric Functions

In the context of a right angle triangle, trigonometric functions play a crucial role in determining unknown angles or sides. In our problem, we're using the tangent function. The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. This can be written as:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} \]
These trigonometric functions also include sine and cosine, which relate to different pairs of a triangle's sides.

• Sine: \( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Cosine: \( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Understanding these functions is essential as they form the basis for angle calculations in right triangles.

Angle Calculation

Calculating angles in a right angle triangle require understanding the relationships within the triangle. In our example, we were able to find angle \( \angle A \) using the inverse tangent function. This step converts the ratio of the opposite to adjacent side into the angle itself:
\[ \angle A = \tan^{-1} \left( \frac{1.0837}{1.4923} \right) \]
This calculation uses a calculator to retrieve the angle in degrees. Once we have \( \angle A \), calculating \( \angle B \) is straightforward using the angle sum property of triangles:

• Sum of angles in a triangle is \( 180^{\circ} \)
• If one angle is \( 90^{\circ} \), the other two must sum to \( 90^{\circ} \)
• \( \angle B = 90^{\circ} - \angle A \)

Calculating angles this way ensures that we have accounted for the geometry of the triangle accurately.

Right Angle Triangle

The geometry of a right triangle is distinctive because one angle is always \( 90^{\circ} \). In right triangles, the side opposite the right angle is the longest side and is called the hypotenuse. This makes right triangles especially handy in calculations involving trigonometric identities.
A right triangle like \( \triangle ABC \) here, helps us apply trigonometric ratios to find missing angles or sides by leveraging the known parts of the triangle. For example:

• The hypotenuse is always opposite the right angle.
• The remaining sides are referred to as the adjacent and opposite sides relative to the angle of interest.
• Angles in the triangle will adhere to the angle sum property, forming part of the path to solving unknown measures.

This clear setup is what permits use of trigonometric ratios efficiently.

Geometry

Geometry is the branch of mathematics dealing with the properties and relations of points, lines, surfaces, and solids. In our exercise, we explore the geometric nature of triangles, focusing on a right triangle.
Geometry offers us principles like the angle sum property, which is crucial for validating calculations. The triangle's configuration and measurement precision allow us to utilize trigonometric functions to solve complex problems.

• Geometric principles ensure that our mathematical models are both accurate and reliable.
• Visualization is often a helpful technique in geometry to better understand and solve problems.
• Using the properties of shapes like triangles can simplify even seemingly complicated tasks.

This harmony between geometry and trigonometry forms a powerful toolset for understanding and solving problems efficiently.