Problem 32 - Trigonometry

Question

$$ \text { Solve each of the following triangles. } $$ Distance Between Two Ships Two ships leave a harbor entrance at the same time. The first ship is traveling at a constant 18 miles per hour, while the second is traveling at a constant 22 miles per hour. If the angle between their courses is \(123^{\circ}\), how far apart are they after 30 minutes?

Step-by-Step Solution

Answer

The ships are about 17.6 miles apart after 30 minutes.

Step 1: Convert Time

Since the time given is 30 minutes, convert it to hours because the speed is given in miles per hour. Divide the minutes by 60: \[ 30 \text{ minutes} = \frac{30}{60} = 0.5 \text{ hours} \]

Step 2: Calculate Distances Traveled

Calculate the distance each ship travels in 0.5 hours using the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \]- For the first ship: \[ \text{Distance}_1 = 18 \times 0.5 = 9 \text{ miles} \]- For the second ship: \[ \text{Distance}_2 = 22 \times 0.5 = 11 \text{ miles} \]

Step 3: Apply the Law of Cosines

Use the Law of Cosines to find the distance between the ships after 0.5 hours. The Law of Cosines states:\[ c^2 = a^2 + b^2 - 2ab \cos C \]where \(a\) and \(b\) are the distances traveled by the ships and \(C\) is the angle between their paths.Substitute \(a = 9\), \(b = 11\), and \(C = 123^{\circ}\):\[ c^2 = 9^2 + 11^2 - 2 \times 9 \times 11 \times \cos(123^{\circ}) \]

Step 4: Calculate Using Trigonometric Values

Calculate \( \cos(123^{\circ}) \) using a calculator. \[ \cos(123^{\circ}) \approx -0.5446 \]

Step 5: Simplify the Equation

Substitute the cosine value into the Law of Cosines and simplify:\[ c^2 = 81 + 121 + 107.7124 \]\[ c^2 = 309.7124 \]

Step 6: Find the Distance

Find \(c\) by taking the square root of \(c^2\):\[ c = \sqrt{309.7124} \approx 17.6 \text{ miles} \]

Step 7: Finalize the Calculation

The distance between the two ships after 30 minutes is approximately 17.6 miles.

Key Concepts

Law of Cosines

The Law of Cosines is a fundamental principle in trigonometry, especially useful for solving triangles when certain information is given. In our exercise, we deal with two ships traveling at different speeds, with a known angle between their paths. The Law of Cosines helps us find the distance between them. The formula \[ c^2 = a^2 + b^2 - 2ab \cos C \] extends the Pythagorean theorem to accommodate angles other than the right angle. Here, \( a \) and \( b \) are the sides, and \( C \) is the included angle.When applying the Law of Cosines:

• Identify the given sides and the included angle.
• Substitute these values into the formula.
• Calculate the cosine of the angle, which can be negative for obtuse angles as we see here with \(123^{\circ}\).

By solving for \( c \), which represents the distance between the ships, we derive that it is approximately 17.6 miles. This showcases the practical application of the Law of Cosines in real-world scenarios where paths diverge.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are vital in understanding angles and lengths in various mathematical problems. In this exercise, we primarily focus on the cosine function.The cosine of an angle \( C \) in a triangle is defined as:

• In a right triangle, it's the ratio of the adjacent side to the hypotenuse.
• For our purposes, it is used in the Law of Cosines for non-right triangles.

Calculating the value of \( \cos(123^{\circ}) \) involves recognizing that it's negative because the angle is greater than 90 degrees. Using a calculator, we roughly determine that \( \cos(123^{\circ}) \approx -0.5446 \). This value is crucial, as it directly impacts the computation:

• It determines how much the angle decreases the distance created by the two paths.
• The negative value of cosine corresponds to the obtuse nature of the angle, affecting the formula result.

Understanding these trigonometric principles is essential in applying them correctly in various mathematical and real-life problems.

Problem Solving Steps

Problem-solving in trigonometry not only requires knowing formulas but also how to approach the problem logically. Here's how the exercise is broken down:

• Convert given units to compatible ones (minutes to hours) to keep consistency with other values.
• Calculate distances traveled by each ship using the speed-time relationship: \( \text{Distance} = \text{Speed} \times \text{Time} \).
• Apply the Law of Cosines with the calculated distances and given angle to find the required distance between ships.

Each step necessitates a clear understanding of the mathematical concepts involved: converting time, handling trigonometric identities, and correctly applying formulas. Further, this structured approach ensures that you systematically and accurately solve the problem. By breaking the problem down into these stages:

• You minimize mistakes and ensure each aspect is considered correctly.
• Overall, it provides a comprehensive solution that bridges mathematics with practical applications.