6.6 Applications of
Trigonometry
Right triangle trigonometry can be used to solve very practical problems. When solving these problems, always make a drawing showing the segments and angles whose measures are given, and the segments and angles whose measures are to be found. Then, in a right triangle, use the proper trigonometric ratios that relate the measures to be found.
For some applications of trigonometry, you need to know the meaning of the following angles.
- Definition:
When an observer sights upward from the horizon, the angle the line of
sight makes with the horizontal is called the angle of elevation.
When an observer sights downward from the horizon, the angle the line of sight makes with the horizontal is called the angle of depression.
The following figures show two possible scenarios involving angles of elevation and depression.
Example: A person whose eyes are 5 feet above the ground is looking at a large mural painted on the wall. The top of the mural is 20 feet above the ground, while the bottom of the mural is 3 feet above the ground. The person is standing 25 feet away from the wall. What are the angle of elevation to the top of the mural and the angle of depression to the bottom of the mural?
Solution:
The situation is pictured to the right. In right $\triangle ABC$,
we have $$\tan \angle ABD=\frac{15}{12}=0.6.$$ Using the table of
trigonometric ratios "backwards" again, we see that the angle of
elevation to the top of the mural is $31^\circ$. In the same way,
in right $\triangle CBD$, we have $$\tan \angle
CBD=\frac{2}{25}=0.08.$$ That means that the angle of depression
to the bottom of the mural is a little under $5^\circ$ (with a
calculator, we find that the angle is $4.6^\circ$). |