Right Triangle Trigonometry

Question

From the observation deck of the lighthouse at Sasquatch Point 51 feet above the surface of Lake Ippizuti, a lifeguard spots a boat out on the lake sailing directly toward the light house. The first sighting had a angle of depression of 8º and the second sighting had an angle of depression of 26.1º. How far had the boat traveled between the sightings?

Answer 1

a/sinα=b/sinβ=c Sin of a rt angle=1.

51/sin8°=366.45=b/sin82°=b/.99; b=362.88

51/sin26.1°=115.925=b/sin63.9°; b=104.104

362.88-104.104=258.78'.

Answer 2

The first sighting line is at 8° depression angle. Draw a line parallel with the lake surface through the observation deck. Opposite angles of parallel lines are equal, so the angle of elevation from the boat is

also 8°. The line of sight from the boat to the deck forms a right triangle with the light house.

Let d₁ be the distance from the boat to the light house base and d₂ be the second sighting distance.

tan(8°) = opp/adjacent = 51/d₁ --> d₁ = 51/tan(8°) = 362.88 feet.

Similarly, for the second sighting, d₂ = 51/tan(26.1°) = 104.1 feet

Distance traveled between sightings = 362.88 - 104.1 = 258.78 feet