Bearing and distance

Question

Two ships A and B leave the same harbour at the same time in the directions 152° and 142° respectively. What is the bearing of A from B at this time when A travelled 10km and B travelled 12km.

Answer 1

There may be an easier way to do this, but here is one way.

Draw a figure!

The point of origin, ship A and ship B form a triangle which has one side 10, one side 12 and a 10° angle between them.

Use the law of cosines to determine the distance between A and B.

Then use the law of sines to get angle B.

Look at your figure to see that the bearing you want is 28° less than angle B.

You can also do this another way:

Point A is (10 cos 142,10 sin 142) and Point B is (12 cos 152,12 cos 152)

The bearing you want is arctan of the slope of the line between these two points