Problem 27 - College Trigonometry

Question

A hiker starts walking due west from Sasquatch Point and gets to the Chupacabra Trailhead before she realizes that she hasn't reset her pedometer. From the Chupacabra Trailhead she hikes for 5 miles along a bearing of \(\mathrm{N} 53^{\circ} \mathrm{W}\) which brings her to the Muffin Ridge Observatory. From there, she knows a bearing of \(\mathrm{S} 65^{\circ} \mathrm{E}\) will take her straight back to Sasquatch Point. How far will she have to walk to get from the Muffin Ridge Observatory to Sasquach Point? What is the distance between Sasquatch Point and the Chupacabra Trailhead?

Step-by-Step Solution

Answer

SM ≈ 4.0 miles; SC ≈ 9.4 miles.

Step 1: Define the Problem and Represent the Triangle

Visualize the scenario as a triangle where Sasquatch Point (S), Chupacabra Trailhead (C), and Muffin Ridge Observatory (M) are the vertices. We need to find the distances SC (pedometer reset point to trailhead) and MS (observatory to Sasquatch Point).

Step 2: Establish Bearing Angles

Translate hiking directions into triangle angles: N 53° W implies an internal triangle angle at C of 90° - 53° = 37°. S 65° E used at M implies an internal triangle angle at M of 180° - 65° = 115° because it refers to the angle away from north.

Step 3: Use the Law of Sines

Apply the Law of Sines: \(\frac{AM}{\sin(37°)} = \frac{SM}{\sin(53°)} = \frac{5}{\sin(115°)}\). This allows us to find SM.

Step 4: Solve for SM using the Law of Sines

Calculate SM by \(SM = \frac{\sin(53°) \times 5}{\sin(115°)}.\) Compute using sines of the specified angles.

Step 5: Determine the Distance SC

Using angle relationships in triangle, find angle S: it's 180° - 37° - 115° = 28°. Now use the Law of Sines to find SC: \(\frac{SC}{\sin(115°)} = \frac{5}{\sin(28°)}.\) Solve for SC: \(SC = \frac{5 \times \sin(115°)}{\sin(28°)} \).

Step 6: Calculate Final Answers

Compute the equations from Steps 4 and 5 to obtain the numerical distances for SM and SC.

Key Concepts

Law of Sines

The Law of Sines is an incredibly useful tool when dealing with oblique triangles, which are triangles that do not have a right angle. It helps us relate the sides and angles of such triangles. This law is often written as:

• \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

where \(a\), \(b\), and \(c\) are the lengths of the sides, while \(A\), \(B\), and \(C\) are the corresponding opposite angles.

To use the Law of Sines, you must know at least one angle-side pair. In our problem, knowing that the hiker moved a specific distance (5 miles) along a bearing helps set the framework for using this formula.

As shown in the exercise, the Law of Sines helps us solve for distances that are not directly measurable by relating them to known distances and angles. In this case, it assists in finding the distances between the Muffin Ridge Observatory and Sasquatch Point as well as between Sasquatch Point and the Chupacabra Trailhead.

Bearing Angles

Bearing angles are an essential concept in navigation, providing directions relative to compass points. They are typically expressed in degrees, starting from north in a clockwise direction. Understanding and converting these angles is crucial, especially when they form parts of geometric problems involving triangles.

In our scenario, the hiker used bearings to navigate between points. The bearing

• \(\text{N} 53^{\circ} \text{W}\) was converted to create an internal angle within the triangle, calculated as \(90^{\circ} - 53^{\circ} = 37^{\circ}\).
• The bearing \(\text{S} 65^{\circ} \text{E}\), moving from Muffin Ridge Observatory back to Sasquatch Point, gives an internal angle of \(180^{\circ} - 65^{\circ} = 115^{\circ}\).

These calculated angles are critical in determining other unknowns within the triangle through trigonometric laws like the Law of Sines. This process demonstrates how bearing angles are not only used for navigation but also play a significant role in solving triangular geometric problems.

Distance Calculation

Calculating unknown distances is a frequent task in trigonometry, particularly within triangular configurations where direct measurements are challenging. In our exercise, the goal was to find the distances between key points in a hiking scenario.

By using the Law of Sines, once you have the necessary angles and at least one side, it's straightforward to find the remaining sides. The process generally follows these steps:

• Identify known distances and angles.
• Calculate the unknown angles if necessary using the sum of angles in a triangle: \(180^{\circ}\) total.
• Apply the Law of Sines to establish a relationship between known and unknown sides.
• Use the sine values of the relevant angles to solve for the desired length.

Using these principles, the hiker's distances back to Sasquatch Point were determined by plugging in the known values into the equations derived from the Law of Sines. Thus, trigonometry becomes a powerful tool to unravel complex distance problems.

Triangle Geometry

Triangle geometry is fundamental to numerous mathematical applications, especially in navigation and surveying. In this exercise, the hikers' path forms a triangle where each side represents a route segment and each vertex a location they wish to reach. Understanding how to calculate different properties of this triangle is essential.

One of the key aspects of triangle geometry is recognizing and using angle relationships:

• Triangles have a total internal angle sum of \(180^{\circ}\).
• Each internal angle can be used in conjunction with known side lengths to determine unknown lengths.

Given the problem's parameters, these principles allow for the calculation of unknown side lengths. After establishing all the internal angles through bearing conversions, solving for the triangle's side lengths becomes a routine application of the Law of Sines.

This illustration of triangle geometry helps enhance our understanding of how trigonometric relations can be applied to real-world scenarios, such as determining hiking paths and distances from bearings and route descriptions.