Problem 24 - College Trigonometry

Question

The SS Bigfoot leaves a harbor bound for Nessie Island which is 300 miles away at a bearing of \(\mathrm{N} 32^{\circ} \mathrm{E}\). A storm moves in and after 100 miles, the captain of the Bigfoot finds he has drifted off course. If his bearing to the harbor is now \(\mathrm{S} 70^{\circ} \mathrm{W}\), how far is the SS Bigfoot from Nessie Island? Round your answer to the nearest hundredth of a mile. What course should the captain set to head to the island? Round your angle to the nearest tenth of a degree.

Step-by-Step Solution

Answer

The SS Bigfoot is approximately 66.34 miles from Nessie Island and should head at \(N 12^{\circ} E\) to reach it.

Step 1: Visualize the Scenario

Imagine a triangle where the harbor is point H, Nessie Island is point N, and the current position of the SS Bigfoot is point B. The original course was 300 miles on a bearing of \(N 32^{\circ} E\) from H to N. After traveling 100 miles, the captain finds out that the bearing to the harbor is \(S 70^{\circ} W\).

Step 2: Calculate Internal Angles

Since the ship is now at point B with the re-calculated bearing to H as \(S 70^{\circ} W\), this means the internal angle at point B (angle HBN) is \(180^{\circ} - 70^{\circ} = 110^{\circ}\). We know angle HNB at N: \(360^{\circ} - (90^{\circ} - 32^{\circ}) = 58^{\circ}\). Using the fact that the sum of angles in a triangle is \(180^{\circ}\), we find angle NHB as \(180^{\circ} - 58^{\circ} - 110^{\circ} = 12^{\circ}\).

Step 3: Apply the Law of Sines

Use the Law of Sines to find the distance from point B to Nessie Island, denoted as \(BN\). The Law of Sines is given by \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Using \(HN = 300\), \(\frac{300}{\sin 110^{\circ}} = \frac{BN}{\sin 12^{\circ}}\). Solve for \(BN\).\[BN = \frac{300 \times \sin 12^{\circ}}{\sin 110^{\circ}}\]Calculate \(BN\) using approximate sine values: \(\sin 12^{\circ} \approx 0.2079\) and \(\sin 110^{\circ} \approx 0.9397\). Substitute these in to get \(BN \approx \frac{300 \times 0.2079}{0.9397} \approx 66.34\).

Step 4: Determine the Course to Nessie Island

The angle \(\angle HBN = 110^{\circ}\) indicates the current line is off by \(110^{\circ} - 90^{\circ} = 20^{\circ}\) off the north-south line towards the west. The new course bearing from the current position towards Nessie Island involves moving towards \(N 32^{\circ} E\) from this 70-degree bearing correction. Set the new course to \(32 - (110 - 90) = 12^{\circ}\), ensuring the SS Bigfoot heads \(N 12^{\circ} E\) toward Nessie Island.

Key Concepts

Bearing Calculations

Understanding navigational trigonometry starts with grasping the concept of bearing. Bearings are used in navigation to describe the direction one point is in relation to another, typically specified in degrees. For example, a bearing of \(N 32^{\circ} E\) means you start facing north and then turn 32 degrees towards the east. It helps to imagine a compass with north at 0 degrees, east at 90 degrees, south at 180, and west at 270 degrees.

Bearings are crucial in navigation to ensure ships and planes reach their destinations accurately. In the case of the SS Bigfoot, the captain's original bearing was \(N 32^{\circ} E\). After the storm, calculating the new bearing of \(S 70^{\circ} W\) required understanding how to translate this direction into internal angles for further triangulation. The process is not just about pointing in a direction but aligning angles to establish precise routes.

Law of Sines

The Law of Sines is a vital tool when dealing with triangles that are not right-angled. This law relates the lengths of sides of a triangle to the sines of their opposite angles, expressed as \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). This formula can help us find unknown sides or angles when some elements are known.

In our example with the SS Bigfoot, after determining the internal angles of the triangle, we applied the Law of Sines to find the distance from point B to Nessie Island, denoted as \(BN\). Knowing the distance of 300 miles from the harbor to Nessie Island and the angle at the ship's current position enabled us to solve for the unknown distance \(BN\).

This method is particularly useful in navigation because weather conditions or unforeseen obstacles can cause deviations from the planned path. The Law of Sines provides a mathematical way to re-calculate and realign navigational strategies.

Triangle Angle Sum

Recognizing that the sum of the angles in any triangle is 180 degrees is a fundamental concept in trigonometry and crucial for solving navigation problems. This principle, known as the Triangle Angle Sum, aids in calculating unknown angles when only some are known.

When the SS Bigfoot's position was off course, determining the internal angles at each point of our imagined triangle—harbor, Nessie Island, and the ship's current location—was necessary for recalibration. For instance, knowing two angles allowed calculation of the third by subtracting their sum from 180 degrees.

Applying the triangle angle sum helps sailors and captains verify bearings and angles, ensuring navigational routes are accurate, especially when adjusting for drift or weather-related course changes. It’s a simple yet indispensable calculation in the vast toolkit of navigation.