Problem 13 - 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. \(a=12\) yards; \(c=37\) yards

Step-by-Step Solution

Answer

\(b = 35\) yards; \(A = 71\degree 56′\); \(B = 18\degree 4′\).

Step 1: Identify the known values

We know side \(a = 12\) yards (adjacent to angle \(A\)) and side \(c = 37\) yards (the hypotenuse). We want to find side \(b\), angle \(A\), and angle \(B\).

Step 2: Calculate side b using the Pythagorean theorem

Using the Pythagorean theorem, \(a^2 + b^2 = c^2\). Substitute the known values: \(12^2 + b^2 = 37^2\). Simplify to find \(b\):\[b^2 = 37^2 - 12^2 = 1369 - 144 = 1225\]\[b = \sqrt{1225} = 35 \, \text{yards}\]

Step 3: Calculate angle A using trigonometry

Use the cosine function: \(\cos(A) = \frac{a}{c} = \frac{12}{37}\). Calculate \(A\):\[A = \cos^{-1}\left(\frac{12}{37}\right) \approx 71.939\degree\]Convert to degrees and minutes: 71° 56′.

Step 4: Calculate angle B as a complement of angle A

Since the sum of angles in a triangle is \(180\degree\) and \(C = 90\degree\), we have:\[A + B = 90\degree\]\[B = 90\degree - A = 90\degree - 71.939\degree = 18.061\degree\]Convert 18.061° to degrees and minutes: 18° 4′.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a key concept in right triangle trigonometry. This theorem 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 lengths of the other two sides. It can be expressed with the equation:

• \(a^2 + b^2 = c^2\)

where \(a\) and \(b\) are the lengths of the legs of the triangle, and \(c\) is the length of the hypotenuse.
For example, given \(a = 12\) yards and \(c = 37\) yards, you can find the missing side \(b\) by rearranging the equation to solve for \(b\):

• \(b^2 = c^2 - a^2 = 37^2 - 12^2\)
• \(b = \sqrt{1225} = 35\) yards

Understanding this theorem allows you to find missing side lengths, which is crucial for solving right triangle problems.

Trigonometric Functions

Trigonometric functions are used to find angles or missing sides of right triangles when one side length and one non-right angle are known. The main trigonometric functions are sine, cosine, and tangent.

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

In the given problem, to find angle \(A\), we use the cosine function since we have the lengths of the adjacent side \(a\) and the hypotenuse \(c\):

• \(\cos(A) = \frac{12}{37}\)
• \(A = \cos^{-1}\left(\frac{12}{37}\right) \approx 71.939\degree\)

Trigonometric functions are essential tools in determining unknown angle measures in right triangles.

Degrees and Minutes

Degrees and Minutes are units used to measure angles. Degrees are the larger unit, and each degree is divided into 60 minutes.

• 1 degree = 60 minutes

This measurement system is similar to how we think about time; for example, 0.5 degrees is equivalent to 30 minutes.
In our problem, angle \(A\) was calculated to be approximately 71.939 degrees. To convert the decimal degrees into degrees and minutes, follow these steps:

• Take the whole number part (71 degrees).
• Multiply the decimal part (0.939) by 60 to get minutes: \(0.939 \times 60 \approx 56\) minutes.
• Thus, angle \(A = 71\degree 56′\).
• For angle \(B = 18.061\degree\), we perform a similar conversion to get \(18\degree 4′\).

This approach ensures precision and is widely used in navigation and surveying.