Problem 39 - Precalculus: A Concise Course

Question

A ship is 45 miles east and 30 miles south of port. The captain wants to sail directly to port. What bearing should be taken?

Step-by-Step Solution

Answer

The bearing that should be taken is 124°

Step 1: Draw and Label the Diagram

Firstly, sketch a right-angle triangle representing the ship's position with the port at the origin, the x-axis pointing east and the y-axis pointing south. The hypotenuse of the triangle will represent the path that the ship wants to take directly to port. Therefore, the ship is at the point (45,-30). Secondly, label the sides of the triangle. The eastward distance (45 miles) is the adjacent side and the southward distance (30 miles) is the opposite side.

Step 2: Find the Angle Using Trigonometric Ratios

We can use tan θ = opposite / adjacent to calculate the angle from the east. Using the given opposite and adjacent sides: tan θ = -30/45 = -0.67. Use inverse tangent function to find angle θ. Choosing the appropriate function on your calculator, we find that θ = arctan (-0.67), which gives a negative angle (about -34 degrees), because our axes are flipped, given we are measuring South of the East, not North as usual.

Step 3: Convert Angle to Bearing

Finally, convert the negative angle from the last step to a bearing. Bearing is always a positive measurement clockwise from the North. Since our angle was measured from the East, subtract the θ found from 90° (since East is 90° from North) to get the bearing. So the bearing = 90° - (-34°) = 124°