Navigation Problem

Trigonometry · Axiom Academy

LESSON Law of Sines: Ship Navigation Navigate the seas using trigonometry! Discover how sailors use angles and the Law of Sines to find their position and chart courses. You're the navigator on a cargo ship in the Atlantic Ocean. Your GPS just failed! But you can see two lighthouses on the coast and measure the angles between them from your position. The Challenge: You know the lighthouses are 20 nautical miles apart. You need to find how far you are from each one to plot your position! Using your sextant, you measure the angles from your ship. The angle at your position (between the two lighthouses) is 60° . You also measure that the bearing to Lighthouse A creates a 45° angle at that lighthouse. Distance between lighthouses (side c): 20 nautical miles Angle at Lighthouse B: 75° (because 180° - 45° - 60° = 75°) Goal: Find distances a and b using the Law of Sines! The Law of Sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant. Each side divided by the sine of its opposite angle: Side a is opposite angle A Side b is opposite angle B Side c is opposite angle C Applying to the ship navigation: A = 45° (at Lighthouse A) B = 75° (at Lighthouse B) C = 60° (at Ship) c = 20 nm (between lighthouses) Let's find the distance from the ship to Lighthouse B (side a ). We'll use: What is the distance from the ship to Lighthouse A (side b )? Hint: Use the same process with angle B = 75°

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