Let's define the East as the x-axis, and the North as the y-axis
Now we can write the speed of the plane as:
(Vx, Vy)
Such that the total velocity of the plane is 350mph
This means that, if the plane flies at an angle of A degrees from the East, then:
Vx = 350mph*cos(A)
Vy = 350mph*sin(A)
We know that the speed of the wind is 65mph at 43° North from East.
This velocity, (Wx, Wy), can be written in components as:
Wx = 65mph*cos(43°) = 47.54 mph
Wy = 65mph*sin(43°) = 44.33 mph
Now, when the plane flies with the wind, the velocity of the plane will be equal to the velocity of the plane plus the velocity of the wind.
This is:
Velocity = (Vx, Vy) + (Wx, Wy) = (Vx + Wx, Vy + Wy)
The plane wants to keep going to the East, then the y-component must vanish (remember that the y-component is the one associated with the North)
Then we must have that:
Vy + Wy = 0
replacing the correspondent values, we have:
350mph*sin(A) + 44.33 mph = 0
350mph*sin(A) = -44.33 mph
Sin(A) = -44.33 mph/350mph
A = Arcsin(-44.33 mph/350mph) = -7.27°
This means that the plane must fly at -7.27° degrees North of East (Or 7.27° South of East)
That is the direction at which the plane must fly.
