C. Problems in Three Dimensions using Trigonometry

So far the problems you have worked with have been in two-dimensions. It is possible to use trigonometry to solve problems in three-dimensions. This can be useful when trying to determine the height of an inaccessible object. Typically, three-dimensional problems will involve triangles in different planes that are butted together.

Example 1

The observations noted in the diagram were made from the shore. The bottom triangle can be thought of as lying flat on the water. Use the information provided to determine the height of the lighthouse.

It may help to label the diagram.

An angle in right triangle BCD is known, so if one of its side lengths can be determined, the height of the lighthouse, BC, can be determined. CD is a side shared by two triangles and there is enough information to determine the length of CD from triangle ACD.

Now use right triangle BCD to determine the height of the lighthouse, BC.

The height of the lighthouse is approximately 113.5 feet.