Hello, geoff9999!

You used the wrong angles . . .

$\displaystyle \text{From the top of a 135-foot tower, a forest ranger sights two forest fires}$

$\displaystyle \text{on opposite sides of the tower. }\,\text{ If their angles of depression are }42.5^o$

$\displaystyle \text{ and }32.6^o,\,\text{ how far apart are the forest fires?}$

In the diagram, "d" represents "degrees".

Code:

P
W - - - - - - - - * - - - - - - - - - E
42.5d * | * 32.6d
* | *
*47.5d|57.4d*
* | *
* |135 *
* | *
* | *
* | *
Q *-----------------*-----------------* R
: - - - a - - - S - - - b - - - :

The tower is: $\displaystyle PS = 135$ ft.

One fire is at $\displaystyle Q.$ .Let $\displaystyle a \,=\,QS.$

$\displaystyle \angle WPQ = 42.5^o \quad\Rightarrow\quad \angle SPQ = 47.5^o$

The other fire is at $\displaystyle R.$ .Let $\displaystyle b \,=\,SR.$

$\displaystyle \angle EPR = 32.6^o \quad\Rightarrow\quad \angle SPR = 57.4^o$

In right triangle $\displaystyle PSQ\!:\;\tan47.5^o \,=\,\dfrac{a}{135} \quad\Rightarrow\quad a \,=\,135\tan47.5^o \,\approx\,147\text{ ft}$

In right triangle $\displaystyle PSR\!:\;\tan57.4^o \,=\,\dfrac{b}{135} \quad\Rightarrow\quad b \,=\,135\tan57.4^o \,\approx\,211\text{ ft}$

Therefore: .$\displaystyle QR \:=\:a + b \:=\: 147 + 211 \:=\:358\text{ ft}$