Problem 58 - Algebra and Trigonometry

Question

Two fire-lookout stations are 10 miles apart with station B directly east of station A. Both stations spot a fire on a mountain to the north. The bearing from station A to the fire is N39°E (39° east of north). The bearing from station B to the fire is N42°W (42° west of north). How far, to the nearest tenth of a mile, is the fire from station A?

Step-by-Step Solution

Answer

Using the provided formula, the calculation gives a distance of approximately 13 miles from station A to the fire.

Step 1: Identify the triangle

The two stations and the fire form a triangle. We know the distance between the stations as 10 miles (the side opposite to the angle we need to find), and the bearings from both stations to the fire.

Step 2: Calculate the correct angle

The angle between the directions to the fire from both stations is 39° (east of north from station A) plus 42° (west of north from station B), which totals to 81°. This is the angle opposite the known side (10 miles).

Step 3: Use the law of sines

We can apply the law of sines to calculate the distance ‘d’ from station A to the fire. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides of the triangle. Therefore, we have \(\frac{d}{\sin(81°)} = \frac{10}{\sin(39°)}\).

Step 4: Solve for ‘d’

By cross-multiplying and solving for 'd', the equation from last step becomes \(d = \frac{10 \times \sin(81°)}{\sin(39°)}\). Calculate the value to find the answer.

Key Concepts

Understanding the Law of Sines

The Law of Sines is a powerful tool in trigonometry that helps us solve for unknown lengths and angles in non-right triangles. It relates the length of a side of a triangle to the sine of its opposite angle. Specifically, it states that for any triangle with sides denoted as \(a, b, c\) and opposite angles \(A, B, C\), the following relationship holds: $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}.$$
This means that the ratio of side length to the sine of the opposite angle is constant for all sides and angles in a triangle. This property is particularly useful when dealing with non-right triangles where only some sides and angles are known.
By applying the Law of Sines, you can determine unknown side lengths or angles, as was done to find the distance from station A to the fire in our exercise.

Exploring Bearing Angles

Bearing angles are used in navigation and surveying to describe directions. They tell you how far you need to turn from a reference direction, typically north or south, towards east or west, to face a target point. These angles are crucial when determining positions and distances.
An example from our exercise includes bearings given as N39°E and N42°W. Here, "N39°E" indicates turning 39 degrees east from due north, while "N42°W" signifies turning 42 degrees west from north.
Bearings are often used in combination with distance measurements to precisely locate an object or point, which in our case is the fire. Understanding how to interpret these angles allows one to accurately frame problems into triangles and use trigonometric solutions like the Law of Sines to find missing information.

Diving into Triangle Properties

Triangles have several unique properties that make them essential in geometry and trigonometry. A triangle's internal angles always add up to 180 degrees. Additionally, knowing two angles gives you the third angle immediately, which helps when solving problems like the one given in the exercise.
The given problem involves a triangle formed by two fire lookout stations and a fire. The stations and their sight lines to the fire create two angles, with the distance between them known. These bearings help determine the angles inside the triangle that are crucial for calculations.
Also, the Law of Sines and other properties, such as the sum of angles, enable solving for unknown distances within the triangle. Understanding and recognizing these properties allows you to convert real-world problems, like the fire and lookout stations, into mathematical models that can be solved methodically.