Hello, Judi!

What do you mean by "use cosine"?

Radar stations $\displaystyle A$ and $\displaystyle B$ are on an east-west line, 8.6 km apart.

Station $\displaystyle A$ detects a plane at $\displaystyle C$, on a bearing of 53°.

Station $\displaystyle B$ simultaneously detects the same plane, on a bearing of 323°.

Find the distance from $\displaystyle B$ to $\displaystyle C.$

Why can I not use cosine to find the length? Code:

C
*
: * * :
: * *37°:
: * * :
: 53° * * :
: * 37° 53° *:
* - - - - - - - - - - - *
A 8.6 B

If you mean "the Law of Cosines", we can't.

. . The Law of Cosines requires *two* sides ... and we have only one.

If you note that $\displaystyle \angle C = 90^o$, you *can* use cosine.

.$\displaystyle \cos B \:=\:\frac{BC}{AB} \quad\Rightarrow\quad BC \:=\:8.6\cos53^o \quad\hdots\:\text{etc.}$