Question

a boat leaves a port and sails 16 miles at a bearing of s 20 degrees e. anoter boat leaves the same port and sails 12 miles at a bearing of s 60 degrees w. how far apart are the two boats at this point

Answer 1

Take the port as the origin. The two courses are each measured from south, one 20° east of south and the other 60° west of south, so the angle between the two paths is 20° + 60° = 80°.

Use the law of cosines on the triangle with sides 16 and 12 and included angle 80°:
d^2 = 16^2 + 12^2 − 2·16·12·cos(80°)
d^2 = 400 − 384·cos(80°)

Thus
d ≈ sqrt(400 − 384·cos 80°) ≈ 18.26 miles (about 18.3 mi).