Hello, ccdalamp!
This type of problem can be solved without the Law of Sines,
. . but it requires more work.
A cable to the top of a tower makes an angle of 35° with the level ground.
At a point 100 yards closer to the tower, the angle of elevation to the top
of the tower is 59°. .Estimate the length of the cable. Code:
* D
* * |
* * |
L * * |
* * |t
* * |
* 35° * 59° |
* - - - - - - * - - - - - - *
A 100 B x C
In the diagram: .$\displaystyle AB = 100,\:BC = x,\:CD = t,\:AD = L$
In right triangle $\displaystyle DCB\!:\;\tan59^o \,= \,\frac{t}{x}\quad\Rightarrow\quad t \,= \,x\tan59^o$ . [1]
In right triangle $\displaystyle DCA\!:\;\tan35 \,=\,\frac{t}{x+100}\quad\Rightarrow\quad t \,= \,(x+100)\tan35$ . [2]
Equate [1] and [2]: .$\displaystyle x\tan59^o \:=\:(x+100)\tan35^o$
. . Solve for $\displaystyle x\!:\;\;x \;=\;\frac{100\tan35^o}{\tan59^o-\tan35^o} \;\approx\;72.63$
In right triangle $\displaystyle DCA\!:\;\cos35^o \:=\:\frac{x+100}{L}\quad\Rightarrow\quad L \:=\:\frac{x+100}{\cos35^o}$
Therefore: .$\displaystyle L \;=\;\frac{72.63 + 100}{\cos35^o} \;\approx\;210.74$ yards.