Triangle Trigonometry

Question

A boat on a lake measures an angle of elevation of 30degrees to the top of a cliff. The boat moves 1000ft closer to the cliff and measures an angle of elevation of 60degrees. How high above the water is the top of the cliff?

Answer 1

We form a triangle with the boat before, the boat 1000 ft later, and the top of the cliff. Note that the angles of the triangle are 30°, 120° (since the 60° angle is exterior) and 180° - 30° - 120° = 30°.

Then, we have an isosceles triangle (2 30° angles), so it is 1000 ft from the boat at its second location to the top of the cliff. Then, we have a second triangle (water from the boat to the cliff, the cliff, and from the boat to the top of the cliff. Then, 1000 ft is the hypotenuse, so the height of the cliff is 1000 ft sin 60° = 1000 * √3/2 =

= 500√3 ft.