Question

The angle of depression of a boat at sea from a 100 foot lighthouse is 20 degrees. Find the distance to the boat.

Answer 1

Use the angle of depression to find the angle opposite to the distance to the boat (x):

The angle of depression and the angle opposite to the distance to the boat are complementary angles (sum up to 90º)

`\begin{gathered} \alpha+20=90 \\ \alpha=90-20 \\ \alpha=70 \end{gathered}`

Use the trigonometric ratio of tangent of angle α:

`\begin{gathered} \tan \alpha=\frac{\text{opposite}}{\text{adjacent}} \\ \\ \tan 70=(x)/(100ft) \end{gathered}`

Solve x:

`\begin{gathered} x=100ft\cdot\tan 70 \\ x=274.8ft \end{gathered}`Then, the distance to the boat is 274.8ft