The pilot of an airplane that flies at 650 km/h wishes to travel to a city 1000 km due east. There is a 75 km/h wind from the northeast.
a)What should the plane's heading be?
b) How long will the trip take?
I take it that you've made a sketch of the Air + Wind = Ground velocity vectors ...
let $\displaystyle G$ = groundspeed along a track due east.
$\displaystyle \theta$ = heading angle relative to east
using components ...
$\displaystyle 650\cos{\theta} - 75\cos(45) = G$
$\displaystyle 650\sin{\theta} - 75\sin(45) = 0$
from the second equation ...
$\displaystyle \theta = \arcsin\left[\frac{75\sin(45)}{650}\right] \approx 4.7^\circ$
plane should steer about $\displaystyle 5^\circ$ north of east.
$\displaystyle G \approx 595$ km/hr
... this problem can also be completed using the law of cosines.