Question

Two lighthouses, A and B, are known to be exactly 20 miles apart on a north-south line. A ship’s captain at S measures ∠ASB to be 33ᵒ. A radio operator at B measures ∠ABS to be 52ᵒ. Find the distance from the ship to each lighthouse.

Answer 1

To find the distances from the ship to each lighthouse, one can use the Law of Sines to calculate AS and BS based on the known angles and the distance between the lighthouses.

Step-by-step explanation:

The student's question involves calculating the distances from a ship to two lighthouses using trigonometry. First, we need to recognize that the ship, S, and the two lighthouses, A and B, form a triangle with angles at A (52 degrees), at S (33 degrees), and hence at B (180 - 52 - 33 = 95 degrees), since the sum of angles in a triangle is always 180 degrees. We can use the Law of Sines to find the distances AS and BS:

Using the Law of Sines: \(\frac{AS}{sin(95)} = \frac{20}{sin(33)}\).Solve for AS by cross-multiplying and dividing: AS = \frac{20 \cdot sin(95)}{sin(33)}.Calculate AS using a calculator.Repeat the process for BS: \(\frac{BS}{sin(52)} = \frac{20}{sin(33)}\).Calculate BS using a calculator.

These steps will give us the distances from the ship to each lighthouse. Remember to use the Sine values for the angles in degrees when making the calculations.