Problem 22 - Algebra 2 and Trigonometry

Question

A distress signal from a ship, \(S,\) is received by two coast guard stations located 3.8 miles apart along a straight coastline. From station \(A,\) the signal makes an angle of \(48^{\circ}\) with the coastline and from station \(B\) the signal makes an angle of \(67^{\circ}\) with the coastline. Find, to the nearest tenth of a mile, the distance from the ship to the nearer station.

Step-by-Step Solution

Answer

The ship is approximately 3.2 miles from the nearer station (station B).

Step 1: Identify Known Quantities

We know that station \( A \) and station \( B \) are 3.8 miles apart. The angles given from the distress signal to the coastline are \( 48^{\circ} \) and \( 67^{\circ} \), respectively. These angles form a triangle with the stations and the location of the distress signal \( S \).

Step 2: Determine the Type of Triangle

The triangle formed is a non-right triangle, as it has two given angles and a side, which is useful for applying the Law of Sines to solve for the unknown distances to point \( S \).

Step 3: Apply the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). For this triangle, let \( a \) be the distance from \( S \) to \( A \) (opposite the angle \( 67^{\circ} \)), \( b \) be the distance from \( S \) to \( B \) (opposite the angle \( 48^{\circ} \)), and \( c = 3.8 \) miles (the distance between \( A \) and \( B \)). The angle at \( S \) is \( C = 180^{\circ} - 48^{\circ} - 67^{\circ} = 65^{\circ} \).

Step 4: Solve for Distance from Ship to Station A

Using the Law of Sines, we set up the equation: \( \frac{a}{\sin 67^{\circ}} = \frac{3.8}{\sin 65^{\circ}} \). Solving for \( a \), we have \[ a = \frac{3.8 \cdot \sin 67^{\circ}}{\sin 65^{\circ}}. \] Calculate \( a \) using a calculator.

Step 5: Solve for Distance from Ship to Station B

Similarly, using \( \frac{b}{\sin 48^{\circ}} = \frac{3.8}{\sin 65^{\circ}} \), solve for \( b \) as follows: \[ b = \frac{3.8 \cdot \sin 48^{\circ}}{\sin 65^{\circ}}. \] Calculate \( b \) using a calculator.

Step 6: Determine the Nearest Station

Compute the distances \( a \) and \( b \) and compare them. Whichever is smaller indicates the closer station to the ship.

Key Concepts

Trigonometry Problem Solving

Trigonometry is an essential tool in solving real-world problems, especially when it comes to unknown distances and angles, like in the exercise you are facing. At its core, trigonometric problem-solving involves using relationships within triangles to find missing sides or angles. In your problem, you're applying the Law of Sines to a triangle formed by two coast guard stations and a distress signal to determine key distances.

To solve such a problem, start by identifying what is known: two angles and a side, between the stations. With this information, a strategy must be developed. In this case, since the triangle is not a right triangle, the Law of Sines is perfect for finding unknown sides based on given angles. When solving for a side of the triangle, we look at the relation:

• For example, the distance from the ship to station A can be given by:
• \[ a = \frac{3.8 \cdot \sin 67^{\circ}}{\sin 65^{\circ}} \]

This approach allows you to systematically find unknowns using known values, emphasizing the need for a structured trigonometric method.

Triangle Properties

Understanding triangle properties helps in effectively applying trigonometric principles. Not all triangles are alike; they can be right triangles, acute triangles, or obtuse triangles. The problem at hand involves an acute triangle, since all angles are less than 90°.

Properties to remember when dealing with triangles:

• The sum of all interior angles always equals 180°.
• An acute triangle has all angles less than 90°, making calculations using sines straightforward.
• In any triangle, the length of one side is always less than the sum of the other two sides.

With the triangle formed by the coast guard stations and the distress signal, we see that the angles are given alongside a side. This configuration is suitable for using the Law of Sines, which leverages angle-side relationships uniquely available because of these properties.

Angle Measurement

Correct angle measurement is critical in accurately solving trigonometric problems. Angles are typically measured in degrees, but radians are also commonly used in advanced mathematics. In this exercise, the angles are measured in degrees, which is standard for most trigonometry problems.

Why proper angle measurement is vital:

• Using incorrect units or wrong angle values can lead to entirely wrong results.
• Understanding that the sum of angles is 180°, you derive the missing angles when necessary, as was done in the solution where angle at S was calculated as:
• \[ C = 180^{\circ} - 48^{\circ} - 67^{\circ} = 65^{\circ} \]

Thus, precise calculations hinge on your ability to correctly measure and interpret angles, ensuring your procedures are accurate and your results are trustworthy.