Question

A plane is flying at a speed and direction of 550 mph, 25 degrees north of west. There is a wind of 28 mph, at an angle of 19 degrees south of the east direction. How fast is the plane actually flying? What direction is the plane traveling?

Answer 1

Here is an easy way to do these.
1) convert all angles to 000N,090E, 180S, 270W


Example: 456@29deg S of E.
= 456@119
then convert to N, E coordinates.

V= 456(Cos119 N + Sin119 E)

Now for your problem:

V= 550(cos295 N +sin295 E)+28(cos109 N+ sin109)
combine N, E vector components:
V= N(550cos295+28cos109)+E(550sin295+28sin109)

do that math and you will have something like this
V= nnnn N + eeeee E

magnitude V= sqrt(nnnn^2+eeee^2)
direction= arctan (eeee/nnnn)
on the angle, be careful, as tangent is a repeating angle, it repeats each 180. So you may have to figure which quadrant.