How far is the ship from the base of the lighthouse?

Question

From the top of a 209-ft lighthouse, the angle of depression to a ship in the ocean is 25°. How far is the ship from the base of the lighthouse? Give your answer to the nearest foot.

Answer 1

construct a right triangle

height = 209

It's a 25 - 65 - 90 degree triangle

height divided by the base = tan25 = opposite side divided by adjacent side

209/b = tan25

base = 209/tan25= distance from the ship to the base of the lighthouse

base = about 209/0.4663= 448.209 feet

= 448 ft. rounded off to nearest foot

or you could use the Law of Sines

b/sin75 = 209/sin25

b = 209sin65/sin25 = 448

side of a triangle divided by sine of the opposite angle is a constant ratio: b/sinB = a/sinA = c/sinC. 209/sin25 = base/sin65

if you work a problem 2 different ways and get the same answer, you have more confidence you have the right answer, more so than just checking the arithmetic