Problem 59 - Algebra and Trigonometry

Question

Lighthouse \(\mathrm{B}\) is 7 miles west of lighthouse \(\mathrm{A}\). A boat leaves A and sails 5 miles. At this time, it is sighted from B. If the bearing of the boat from \(\mathrm{B}\) is \(\mathrm{N} 62^{2} \mathrm{E}\), how far from \(\mathrm{B}\) is the boat? Round to the nearest tenth of a mile.

Step-by-Step Solution

Answer

The boat is approximately 2.1 miles away from lighthouse \( B \).

Step 1: Formulate the Problem

The problem involves three points: lighthouse \( A \), lighthouse \( B \), and the boat. With \( A \) and \( B \) we have the base of the triangle and with information about the distance from \( A \) to the boat, we can determine one side of the triangle. The bearing from \( B \) provides additional information required to solve the problem. Draw these connections on a diagram.

Step 2: Identify Angles

Given that the bearing from \( B \) is \( N62.5^{2} E \), you can deduce that the angle \( \alpha \) between \( B \) and the boat is \( 90 - 62.5 = 27.5^{\circ} \). The triangle is a non-right triangle, hence the law of sines will be applicable here.

Step 3: Use the Law of Sines to Solve

From the law of sines, it is known 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 three sides of the triangle. You already calculated one angle \( \alpha \), already know the opposite side (which is 7 miles) and have the information to calculate the other angle \( \beta = 180 - 90 - 27.5 = 62.5^{\circ} \). Use these details to solve for the distance \( x \) between the boat and lighthouse \( B \). \( x = \frac{5 \cdot \sin(27.5)}{\sin(62.5)} \) , after calculating this expression, round your answer to the nearest tenth of a mile.

Key Concepts

Law of Sines

The Law of Sines is an important concept in trigonometry that helps to solve problems involving non-right triangles. It states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is identical for all three sides. For triangle ABC, this can be expressed as:

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

This law is particularly useful when you know one of the sides and two angles or two sides and one angle of a triangle. In the context of the given problem, once we identified the angle between point B and the boat, we used this law. We had the side opposite the angle at B as 7 miles and found the angle at B to be 27.5 degrees. By applying the Law of Sines, we can find the third side, which is the distance from the boat to lighthouse B. This calculation makes it easy to solve for unknown distances in triangular navigation problems, like the one involving the lighthouses and the boat.

Bearing

The concept of bearing is essential in navigation and is used to describe the direction of one point from another. It is measured in degrees, typically from the north direction. For example, a bearing of N 62.5° E means the direction is 62.5 degrees east of true north.
Bearings are often expressed using the three parts:

• The cardinal direction (N, S, E, W) indicating the starting point.
• The angle in degrees showing the deviation from that starting point.
• The ending cardinal direction indicating the line's final direction.

In the given problem, the bearing of N 62.5° E from lighthouse B gave us the necessary information to determine the angle relevant to our triangle. Bearings can often seem complex, but breaking them down helps understand directions more straightforwardly, especially when visualizing the problem on a diagram.

Triangle

Understanding the basic properties of a triangle is fundamental when solving geometric problems. A triangle is a three-sided polygon, and its angles always sum to 180 degrees. In navigation problems like the one given, knowing different types of triangles and how to identify side lengths and angles helps in setting up the problem correctly.
There are different triangles:

• Equilateral, where all sides and angles are equal.
• Isosceles, with two equal sides and angles.
• Scalene, with all sides and angles of different lengths and degrees.
• Right, having one 90-degree angle.

For non-right triangles, such as the one in this problem, we use methods like the Law of Sines to find unknown distances and angles. By forming a clear image of the triangle in question and knowing the type, solving the problem becomes systematic and logical.

Angle Calculation

Angle calculation is crucial when dealing with any trigonometric or geometric problem. Knowing how to find missing angles using given angles and side lengths leads to determining other essential variables in triangle problems.
Given a bearing, the problem required converting it into an angle that could be used for further calculation. Starting with a bearing of N 62.5° E, we found the corresponding angle in the triangle by subtracting 62.5 from 90, giving an angle of 27.5 degrees.

• Angles in a triangle add up to 180 degrees.
• Use known angles and side lengths to find unknowns through trigonometric ratios.
• Keep consistent units and measurements to avoid misunderstandings.

Recognizing that the total of the angles in a triangle is always 180 degrees, we calculated the other angles to use the Law of Sines. Angle calculation ensures that every triangle-related puzzle has a logical and straightforward solution.