This is a right triangle, so we know one angle already, angle 
is 
.
We know some relationships between the side lengths and the trig functions of angles 
and 
.
The Pythagorean Theorem gives us a relationship between teh side lengths:
In general we would need to worry about a 
when solving such an equation, but in these examples we are looking for lengths, so we only want the positive solution.
We also know the following relationships between trig functions and side lengths:
As a consequence of these relations we are able to solve for all the side lengths and all the values of the trig functions if we are given either
Here's an example of situation 1:
Suppose we know that 
and 
.
For angle 
, the adjacent side is 
, the opposite side is 
, and the hypoteneus is 
.
We know that
Check that
For angle 
, the adjacent side is 
, the opposite side is 
, and the hypoteneus is 
.
Notice that when we compute the trig functions with repect to angle 
instead of angle 
the 
and 
switch and the 
becomes the reciprocal.
Understanding this relationship saves work, there is no need to do separate calculations for the trig functions of angle 
.
Here's an example of situation 2:
Suppose 
and 
.
Now we know 
and 
, so the rest of the argument is as before.
Again, a good thing to do is check that 
.
Now suppose that 
and 
. Can you find the rest of the values?
Suppose you are standing 50 yards from a flagpole, and the line to the top of the flagpole makes an angle of
from horizontal.
How tall is the flagpole?
The height of the flagpole will be the opposite side of a right triangle. The adjacent side of the triangle runs from your feet to the base of the flagpole and is 50 yards long.
Make sure your calculator is in degrees mode to calculate
.
Here's another: Suppose you stand 150 ft from a building with a tall smokestack. The angle of elevation to the top of the building is
.
The angle between the top of the building and the top of the smokestack is 
.
How tall is the smokestack alone, not including the height of the building?
First find the height of the building. With respect to the
angle, we are looking for the opposite side and we know that the adjacent side is 150 ft, and that
so,
We can do the same calculation using the sum of the two angles to find the total height from the ground to the top of the somkestack.
So the height of the smokestack alone is
The elevation of Mt. Olympus is 9,026 ft. The elevation at the surface of the Great Salt Lake is 4200 ft. If you stand at the shore of the lake and observe that the peak of Mt. Olympus has an angle of elevation of
what is your apparent (ignoring curvature of the Earth) horizontal distance from the peak of Mt. Olympus, in miles?
This time we know the opposite side and want the adjacent side. The opposite side is the apparent height of Mt. Olympus,
ft.
