Trigonometry word problem? Two ships have radio equipment with a range of 200 km.One is 155km North42 degrees 40 minutes east and the other is 165 km north 45 degrees 10 minutes west of a shore station. Can the two ships communicate directly? 
How would I solve this problem I know I have to make a triangle but I am not sure how the triangle would look like. 
 A: We will assume that the Earth is flat, and that we are not too far North.  Draw a picture. Let the share station be at $C$. For the first ship, go $42^\circ40'$ East from due North, and travel $155$ km. Call the resulting point $A$. So we face North from $C$, turn $42^\circ40'$ clockwise, and sail $155$ km.   
For the second, go $45^\circ10'$ West from due North, and travel $165$ km. Call the resulting point $B$.
Then $\triangle ABC$ has $CA=155$, $CB=165$, and $\angle C=42^\circ40'+45^\circ10'=87^\circ 50'$.
By the Cosine Law,
$$(AB)^2=155^2+165^2-2(155)(165)\cos C.$$
Calculate. It turns out that the distance is about $222$ km.
Remark: We can proceed more informally without the Cosine Law, and with a little crossing of the fingers. Note that $\angle C$ is almost a right angle. If we pretend it is a right angle, we can use the Pythagorean Theorem to estimate the distance. That gives $226$ km. Not very different from $222$.
A: With the shore station as the common vertex there should be two right angled triangles, one with angle 42 degrees 40 minutes and other with angle 45 degrees 10 minutes to yhe otjer side since it is west.
