Hello, jul07010!

I agree with u2_wa: we don't need the Law of Sines . . .

A point on the ground is 200 feet from a water tower.

The angle of elevation to the top of the tower is 18°.

The angle of elevation to the bottom of the tower is 15°.

How tall is the water tower?

(i think the water tower is on a hill?) yes! Code:

B *
| *
| *
h | *
| *
| *
C * *
| * *
| * *
y | 200 * 3° *
| * *
| 15° * *
D * - - - - - - - - - - - * A
x

The height of the tower is: $\displaystyle h = BC$

Let $\displaystyle y = CD,\;x = DA.$

$\displaystyle AC$ is the ground: $\displaystyle AC = 200.$

$\displaystyle \angle CAD = 15^o,\;\angle BAD = 18^o$

In right triangle $\displaystyle CDA:\;\begin{array}{ccccccc}\sin15^o \:=\:\frac{y}{200} &\Rightarrow& y \:=\:200\sin15^o & {\color{blue}[1]}\\

\cos15^o \:=\:\frac{x}{200} & \Rightarrow & x \:=\:200\cos15^o & {\color{blue}[2]} \end{array}$

In right triangle $\displaystyle BDA\!:\;\tan18^o \:=\:\frac{h+y}{x} \quad\Rightarrow\quad h \;=\;x\tan18^o - y $ .[3]

Substitute [1] and [2] into [3]: .$\displaystyle h \;=\;(200\cos15^o)\tan18^o - 200\sin15^o $

Therefore: .$\displaystyle h \;\approx\;11$ ft.