Example 1

Ava has hiked to the top of a hill and can see a ruined cabin below her in the distance. Using her map, she estimates that she is a horizontal distance of 2.3 km from the cabin. She then estimates that the angle of depression to the cabin is 15°. How much higher, to the nearest tenth of a kilometre, is Ava than the cabin?

Begin by sketching a diagram to represent the problem.

Next, make a plan. The goal is to determine h. You now know two methods for determining an unknown side length in a right triangle:

  • Use the Pythagorean theorem when two other side lengths of the same right triangle are known.

  • Use a trigonometric ratio when one side length and one angle measure of the same right triangle are known.

If you can determine either another side length or another angle measure in the right triangle, you will be able to solve the problem. The 15° angle and the other angle, a, at the top of the hill form a right angle, so they must add to 90°.

There is now enough information to determine h using the tangent ratio.

Ava is approximately 0.6 km higher than the cabin.