Trig
posted by Alexis on .
A billboard rests on top of a base. From a point level with the bottom of the base, the angle of elevation of the bottom of the billboard is 26 degrees and the top of the billboard is 31 degrees. From a point 21 meters further away, the angle of elevation to the top of the billboard is 17 degrees. Find the height of the billboard.

Let's make a diagram.
Call the top of the billboard A and the bottom of the billboard E, and the point below AE we will call C, where C is at ground level.
Label D as the first point of observation, and B as the second point of observation, so that BD = 21
Let's look at triangle ABD.
Angle BDA = 18031 = 149º
then angle BAD = 14º
and angle B = 17º
So by the sine law,
AD/sin17 = 21/sin14
We can find AD.
Now let's look at triangle ADC which is rightangled.
cos 31 = DC/AD, thus we can find DC
also sin 31 = AC/AD, giving us AC
and finally in triangle DEC
tan 26 = EC/DC giving us EC
so the billboard's height is AC  EC
Do not round off preliminary answers, good luck. 
Let a be the original horizontal distance from which the billboard is viewed.
Let h be the height of the billboard.
Let b be the height of the base.
b/a = tan 26 = 0.4877
(b+h)/a = tan 31 = 0.6009
(b+h)/(a + 21) = tan 17 = 0.3057
You have to solve three equations in three unknowns (a, b, h) to get h.