Question

a ships navigator determines that the ship is moving through the water at 17.5mi/h with a heading of 26.3 degrees north of east but the ship is actually moving at 19.3 mi/h in a direction of 33.7 degrees north of east. What is the velocity of the current? would this be 17.5cos26.3 and 19.3cos33.7 then find the difference?

Answer 1

you can get the magnitude by law of cosines...draw the figure.
Then to get the heading of the current, use law of sines.

Answer 2

Vs + Vc = 19.3 cos 33.7 i + 19.3 sin 33.7 j

Vs = 17.5 cos 26.3 i + 17.5 sin 26.3 j

so
Vc x = 19.3 cos 33.7 - 17.5 cos 26.3
and
Vc y = 19.3 sin 33.7 - 17.5 sin 26.3

|Vc|^2 = (Vc x)^2 + (Vc y)^2

tan angle N of E = Vc y/ Vc x

of course in the real world navigators measure angles east of north, not north of east, but people who write math books do not operate ships :)