Problem 30 - College Trigonometry

Question

A hiker determines the bearing to a lodge from her current position is \(\mathrm{S} 40^{\circ} \mathrm{W}\). She proceeds to hike 2 miles at a bearing of \(\mathrm{S} 20^{\circ} \mathrm{E}\) at which point she determines the bearing to the lodge is \(\mathrm{S} 75^{\circ} \mathrm{W}\). How far is she from the lodge at this point? Round your answer to the nearest hundredth of a mile.

Step-by-Step Solution

Answer

The hiker is approximately 1.15 miles from the lodge.

Step 1: Understand the Problem

The hiker starts at point A, hikes to point B, and at point B measures the bearing to the lodge at point L again. We started with the bearings: A to L is \( S 40^{\circ} W \), A to B is \( S 20^{\circ} E \), and B to L is \( S 75^{\circ} W \). We need to find the distance BL.

Step 2: Visualize the Bearings with a Diagram

Draw a diagram where A is the origin. The direction \( S 40^{\circ} W \) to L from A will lie in the southwest quadrant making an angle of 40 degrees with the south. The hiker then hikes 2 miles \( S 20^{\circ} E \) to reach B. The direction from B to the lodge L is \( S 75^{\circ} W \).

Step 3: Set Up the Right Triangle

We can form a triangle ABL with sides AB = 2 miles and bearing angles relative to the north-south line. At A, the angle BAL is \( 40^{\circ} + 20^{\circ} = 60^{\circ} \). At B, L is \( 75^{\circ} \) west of south. Add this angle to get the internal angle BLA: \( 180^{\circ} - (20^{\circ} - 75^{\circ}) = 85^{\circ} \). Angle LAB then is \( 180^{\circ} - (60^{\circ} + 85^{\circ}) = 35^{\circ} \).

Step 4: Apply the Law of Sines

Using the Law of Sines, we solve for BL. \[ \frac{AB}{\sin(BLA)} = \frac{BL}{\sin(BAL)} \]Plug the known values:\[ \frac{2}{\sin(85^{\circ})} = \frac{BL}{\sin(35^{\circ})} \]Solving this gives:\[ BL = \frac{2 \times \sin(35^{\circ})}{\sin(85^{\circ})} \approx 1.15 \] miles.

Key Concepts

Bearing Calculations

Bearing calculations help us navigate by determining directions using angles. It's like using a compass! In this problem, a series of bearings help find the distance to a lodge. Bearings are expressed in terms of compass directions, and each bearing tells us how far and in which direction we should go.

• A typical bearing uses the north or south line as a reference. Then, it indicates the clockwise angle to the east or west.
• The hiker's first bearing is \( S 40^{\circ} W \), which means starting from the south, turn 40 degrees towards the west.
• The second bearing when reaching point B is \( S 75^{\circ} W \), signifying a turn of 75 degrees towards the west from the south line.

These movements can be illustrated visually, making it easier to understand the changes in position, crucial in triangulating the hiker's path. Each bearing gives critical angles needed to calculate distances in triangular problems.

Law of Sines

The Law of Sines is a powerful tool in trigonometry, especially for solving triangles that are not right-angled. It's a formula that relates the lengths of a triangle's sides to the sines of its angles. In our exercise, this law is essential to find the distance to the lodge.
With triangle ABL, the important angles and sides are:

• Angle BAL formed by the initial bearing plus the hiker's path: \(40^{\circ} + 20^{\circ} = 60^{\circ}\).
• Angle BLA: \(85^{\circ}\) as the difference from \(180^{\circ}\).
• Side AB initially given as 2 miles.

By applying the Law of Sines, we use this relationship:\[ \frac{AB}{\sin(BLA)} = \frac{BL}{\sin(BAL)} \]This formula allows us to insert known angle measures and side length to determine the unknown distance BL. Calculations show that the hiking distance to the lodge is approximately 1.15 miles.

Triangular Bearing Method

The triangular bearing method combines geometry and trigonometry to determine unknown distances and directions in navigation. It involves plotting a triangle based on angular bearings which link a sequence of points.In this scenario:

• The hiker's path from point A to B forms one leg of our triangle, using both the direct south line bearing and any deviation east or west.
• At point B, another bearing line extends towards point L, creating a new angle influenced by the hiker's earlier path.

This method relies on the fact that any triangle's internal angles add up to \(180^{\circ}\). First, we calculate angles based on the compass bearings. Then we solve the triangle using methods like the Law of Sines to find unknown sides, ensuring we have enough information from our initial bearings and set path lengths.By effectively using the triangular bearing method, we're able to transform challenging navigation questions into straightforward mathematical exercises, neatly guiding the hiker's journey as if following a defined map.