An airplane is flying on a bearing of 80 degrees at 540mph. A wind is blowing with a bearing of 100 degrees at 55mph. Find the component form of the velocity of the airplane.

Answer 1

Mathematicians do not do navigation unfortunately so I suspect they mean that the wind is heading 100 degrees rather than coming from there (the way we navigators express wind vectors)
In that case add the plane velocity vector to the wind velocity vector:

80 degrees at 540mph +100 degrees at 55mph.

by the way 80 is ten north of east
and
100 is 10 south of east or 80 east of south

East component (x) = 540 cos 10 + 55 cos 10

North component (y) = 540 cos 80 - 55 cos 80