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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 37
Chapter 8, Problem 37

The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.

Verified step by step guidance
1
Draw a diagram to visualize the problem: mark the lighthouse as a fixed point, and the two ship positions. The first position has a bearing of N 37° E to the lighthouse, and after sailing 2.5 miles due south, the second position has a bearing of N 25° E to the lighthouse.
Assign variables to the distances from the lighthouse to the ship at the first and second positions, say \(d_1\) and \(d_2\) respectively. The bearings give the angles between the north direction and the line from the ship to the lighthouse.
Using the bearings, express the coordinates of the ship relative to the lighthouse. For the first position, the ship lies along a line making an angle of 37° east of north from the lighthouse. For the second position, after moving 2.5 miles south, the ship lies along a line making an angle of 25° east of north.
Set up equations using trigonometric relationships. For example, use the sine and cosine of the bearing angles to relate the east-west and north-south components of the distances. The vertical displacement between the two ship positions is 2.5 miles south, so use this to relate \(d_1\) and \(d_2\).
Solve the system of equations to find \(d_1\) and \(d_2\), which represent the distances from the lighthouse to the ship at the first and second positions respectively.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Understanding Bearings

Bearings are a way to express direction relative to the north, measured in degrees clockwise. For example, N 37° E means 37 degrees east of due north. Interpreting bearings correctly is essential to set up the problem and visualize the ship's and lighthouse's relative positions.
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Example 2

Using Trigonometric Ratios in Right Triangles

Trigonometric ratios like sine, cosine, and tangent relate angles to side lengths in right triangles. By modeling the ship's positions and bearings as triangles, these ratios help calculate unknown distances between the ship and lighthouse based on given angles and movements.
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Solving Right Triangles with the Pythagorean Theorem

Applying the Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem for any triangle, relating side lengths and included angles. It is useful when two sides and the included angle or two angles and a side are known, enabling calculation of unknown distances in non-right triangles formed by the ship's movement and bearings.
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Intro to Law of Cosines
Related Practice
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