a boat sails 160km from a harbor,P on a bearing of 36 degree to Q and 200km on a bearing of 125 to another habour

Question

What is the distance between P and R and what is the bearing of P from R

Answer 1

Bearings are in relation to north. In the triangle PQR, angle Q=36+180-125=91 degrees.

Using the cosine rule:

PR^2=PQ^2+QR^2-2PQ.QRcosQ=25600+40000-2*32000cos91=66716.95 approx.

Therefore PR=258.3km approx.

We can find other angles in PQR using the sine rule, so sinP/200=sin91/258.3=0.003871 approx.

sinP=200*0.003871=0.7742 approx. From this P=50.73º.

To find the bearing we need to add 36º, the initial bearing of Q from P=86.73.