Problem 43 - Algebra and Trigonometry

Question

A fisherman leaves his home port and heads in the direction \(\mathrm{N} 70^{\circ} \mathrm{W}\) . He travels 30 \(\mathrm{mi}\) and reaches Egg Island. The next day he sails \(\mathrm{N} 10^{\circ} \mathrm{E}\) for 50 \(\mathrm{mi}\) , reaching Forrest Island. (a) Find the distance between the fisherman’s home port and Forrest Island. (b) Find the bearing from Forrest Island back to his home port.

Step-by-Step Solution

Answer

(a) 62.6 mi; (b) 298.3° bearing.

Step 1: Draw the problem

First, visualize and draw a sketch of the problem. Mark the home port as point H, Egg Island as point E, and Forrest Island as F. The path H to E heads N 70° W, and E to F heads N 10° E.

Step 2: Use the Law of Cosines

From the drawing, recognize that triangle HEF is a non-right triangle. Use the Law of Cosines to find the distance HF. The angle HEF is found by calculating the angle between the two paths:\[\angle HEF = 180^{\circ} - 70^{\circ} - 10^{\circ} = 100^{\circ}\]Apply the Law of Cosines:\[HF^2 = HE^2 + EF^2 - 2 \cdot HE \cdot EF \cdot \cos(\angle HEF)\]Substitute the values:\[HF^2 = 30^2 + 50^2 - 2 \cdot 30 \cdot 50 \cdot \cos(100^{\circ})\]

Step 3: Solve for HF

Calculate using the values from the previous step:\[HF^2 = 900 + 2500 - (3000 \times -0.1736) = 3400 + 520.8 \]\[HF^2 = 3920.8\]\[HF = \sqrt{3920.8} \approx 62.6\]Thus, the distance between the home port and Forrest Island is approximately 62.6 miles.

Step 4: Find Bearing Using Law of Sines

To find the bearing from Forrest Island back to the home port, use the Law of Sines. First, find angle EFH:\[\frac{\sin(EFH)}{EF} = \frac{\sin(HEF)}{HF}\]\[\frac{\sin(EFH)}{50} = \frac{\sin(100^{\circ})}{62.6}\]Calculate:\[\sin(EFH) = \frac{50 \times \sin(100^{\circ})}{62.6} \approx 0.7864\]\[EFH \approx \arcsin(0.7864) \approx 51.7^{\circ}\]

Step 5: Calculate Bearing

The bearing from Forrest Island to the home port involves combining the found angle and the known path direction north. The bearing is found by adjusting for the triangle's direction:\[\text{Bearing} = 360^{\circ} - (10^{\circ} + 51.7^{\circ}) = 298.3^{\circ}\]So the bearing from Forrest Island back to the home port is approximately 298.3°.

Key Concepts

Law of Sines

The Law of Sines is a powerful tool in solving problems involving triangles. It is especially helpful when dealing with non-right triangles, which are triangles that do not include a 90-degree angle. Here's how it works:

The Law states that in any triangle, the ratios of the length of a side to the sine of its opposite angle are equal for all three sides. The formula is expressed as follows:

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

where \(a, b, c\) are the lengths of the sides, and \(A, B, C\) are the opposite angles of those sides respectively.

In practice, this means if you know any three of these six pieces of information (at least one must be a side), you can find the other three.

For example, in the problem involving the fisherman, the Law of Sines was used to determine the angle \(EFH\) by substituting known side lengths and the angle \(HEF\)

This calculation is crucial in determining the bearing back to the home port.

bearing calculation

Bearing calculation is crucial in navigation to determine direction. A bearing is an angle measured clockwise from the north direction. It helps navigators describe the direction of one location from another.

Bearings are typically expressed in degrees, ranging from 0 to 360. For example, north is 0 degrees, east is 90 degrees, south is 180 degrees, and west is 270 degrees.

In the fisherman's case, to find the bearing from Forrest Island back to the home port, we combined the internal angles of the path with the cardinal directions. By knowing the angle \(EFH\) was approximately \(51.7^{\circ}\), we could adjust for the path's direction north to find the bearing.

Using this calculation technique, we expressed the return direction as bearing \(298.3^{\circ}\), signifying how one would travel back to the start.

non-right triangles

Non-right triangles are any triangles that do not include a 90-degree angle. These triangles require different methods to solve compared to right triangles, as trigonometric relationships like the Pythagorean theorem do not apply directly.

Instead, we utilize laws such as the Law of Sines and the Law of Cosines, which allow us to solve for unknown sides or angles. These laws help us work with the triangles' oblique nature.

For instance, in the fisherman's situation, the triangle formed by his journey is a non-right triangle. This made directly using simple trigonometric equations impossible. Instead, the Law of Cosines was employed. The triangle's internal angle \(HEF\) was determined, facilitating the calculation of the distance from the home port to Forrest Island.

Non-right triangles also play a significant role in real-world applications like navigation, where precise measurements are vital for reaching a destination accurately.