Hello, robasc!
I believe you have the 34° angle in the wrong place.
1. A pole 60 feet tall is situated at the bottom of a hill that slopes up at an angle of 15.9°.
A guy wire from the top of the pole to the hillside forms an angle of 34° with the top of the pole.
Find the distance from the base of the pole to the guy wire's point of attachment. Code:
A o
| \
|34°\
| \
| \
| \
60 | \
| \ *
| \ C *
| 71.9° o
| 74.1° *
| * 15.9°
B o - - - - - - - - - - - - - - - D
The pole is: $\displaystyle AB = 60$
The ground is $\displaystyle BD.\;\;\angle CBD = 15.9^o \quad\Rightarrow\quad \angle ABC = 74.1^o$
The guy wire is $\displaystyle AC.\;\;\angle BAC = 34^o \quad\Rightarrow\quad \angle ACB = 71.9^o$
Law of Sines: .$\displaystyle \frac{BC}{\sin 34^o} \:=\:\frac{60}{\sin71.9^o} \quad\hdots$ etc.