Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. \(a=76.4\) yards; \(b=39.3\) yards

Step-by-Step Solution

Answer

The right triangle has \( c \approx 85.94 \) yards, \( A \approx 62.0^{\circ} \), and \( B \approx 28.0^{\circ} \).

Step 1: Identify Right Triangle Components

We are given that this is a right triangle with angle \( C = 90^{\circ} \). The sides are labeled as \( a = 76.4 \) yards and \( b = 39.3 \) yards. Our goal is to find angle \( A \), angle \( B \), and the side \( c \), using the provided measurements.

Step 2: Calculate Hypotenuse Using Pythagorean Theorem

To find the length of side \( c \), we use the Pythagorean Theorem: \( c = \sqrt{a^2 + b^2} \). Substituting for \( a \) and \( b \), we have: \( c = \sqrt{76.4^2 + 39.3^2} \). Calculating this gives \[ c \approx \sqrt{5839.36 + 1544.49} = \sqrt{7383.85} \approx 85.94 \] yards.

Step 3: Find Angle A Using Tangent Function

The tangent of angle \( A \) is given by \( \tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} \). So, \( \tan A = \frac{76.4}{39.3} \). Calculating gives \( \tan A \approx 1.944 \). Using the inverse tangent function to find \( A \), we have \( A = \tan^{-1}(1.944) \). Calculating this gives \( A \approx 62.0^{\circ} \).

Step 4: Find Angle B Using Complementary Angle Rule

Since the angles in a triangle sum to \( 180^{\circ} \) and \( C = 90^{\circ} \), angles \( A \) and \( B \) must add up to \( 90^{\circ} \). Thus, \( B = 90^{\circ} - A = 90^{\circ} - 62^{\circ} = 28^{\circ} \). Therefore, \( B \approx 28.0^{\circ} \).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially useful in right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be mathematically expressed as:

• \( c^2 = a^2 + b^2 \)

This relationship helps us calculate the unknown side when the other two sides are known. In our exercise, we used this theorem to find the hypotenuse, \( c \), by substituting \( a = 76.4 \) yards and \( b = 39.3 \) yards into the equation:

• \( c = \sqrt{76.4^2 + 39.3^2} \)

Which helped us calculate \( c \approx 85.94 \) yards. This straightforward calculation is a great reminder of how useful the Pythagorean Theorem is for solving problems involving right triangles.

Trigonometric Functions

Trigonometric functions allow us to find unknown angles and sides in right triangles. The primary functions used here are sine, cosine, and tangent. Specifically, the tangent function is defined as the ratio of the length of the opposite side to the adjacent side for any angle in a right triangle:

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

In solving the exercise, we used the tangent of angle \( A \) to find its measurement:

• \( \tan A = \frac{76.4}{39.3} \)

This calculation yielded \( \tan A \approx 1.944 \). To find the angle \( A \), we used the inverse tangent function, often denoted as \( \tan^{-1} \) or \( \arctan \), which leads us to \( A \approx 62.0^{\circ} \). Understanding and using trigonometric functions in this way is essential for solving and interpreting angle measurements in triangles.

Complementary Angles

In the context of right triangles, complementary angles play a significant role. For any right triangle, the two non-right angles are complementary, meaning their measures add up to \(90^{\circ}\). Given our triangle, where the right angle is \(C = 90^{\circ}\), angles \(A\) and \(B\) must satisfy:

• \( A + B = 90^{\circ} \)

Once we determined \(A\) to be approximately \(62.0^{\circ}\), finding \(B\) was straightforward since:

• \( B = 90^{\circ} - A = 90^{\circ} - 62^{\circ} \)

Thus, \(B \approx 28.0^{\circ}\). Understanding complementary angles helps in easily calculating the remaining angles in situations where one angle is known, truly an efficient technique in elementary geometry.

Angle Measurement

When dealing with angles, precise measurement is crucial to accurate triangle solutions. Angles can be expressed in several ways, such as degrees, minutes, and seconds, or as decimal degrees. In this exercise, we worked with angles in degrees and minutes. This system divides a degree into 60 minutes, and each minute into 60 seconds, similar to the time measurement system. This subdivision helps express angles with greater precision. For example, an angle can be given as \(62^{\circ} 0'\), which implies it's exactly 62 degrees without any fractional degree parts. Understanding these measurements and conversions is fundamental to solving real-world geometric problems accurately. When converting between these representations, it's essential to be familiar with this breakdown to maintain angle measurement accuracy.