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.