I'm working through a book to learn trig on my own and I got stuck with the following. This is the image given and the text in the book reads:
Suppose you are standing an unknown distance away from a cliff of height $h$. You need to know the height $t$ of a tower located on top of the cliff. You know that the angle of elevation of the bottom of the tower is $B$ and the angle of elevation of the top of the tower is $A$. Derive a formula for the height of the tower.
The solution they gave is $t = h\left({{tan A}\over{tan B}}-1\right)$. I don't quite understand how to derive that answer. I think that the height is going to be $d(tanA – tanB)$ where $d$ is the unknown adjacent side. I get stuck after that. The more verbose the answer the better. Thanks!
Best Answer
We have $$\tan A=\frac{t+h}d\iff d=\frac{t+h}{\tan A}$$
Similarly, $$\tan B=\frac hd\iff d=\frac h{\tan B}$$
Compare the two values of $d$