I need to find the adjustment the pilot have to make to keep the plane in the due east intial direction?

Question

A Boeing 727 jet is flying due east at 500 mph in still air. A 70 mph tail wind approaches in the direction 60 degree north of east. What adjustment does a pilot have to make to keep the plane due est initial direction?

Answer 1

you need to combine

-70 (N 30° E) ... {that is the usual heading format..}
with 500 (N (90°-θ) E)

Now...
-70 (N 30° E) = -70 〈 cos 60° , sin 60° 〉 = 〈 -35 , -60.622 〉
500 (N (90°-θ) E) = 〈 500 cosθ , 500 sinθ 〉

thus 500 sin θ = 60.622 ... it is important that the resultant y component is zero... we dont care about the x component...
sin θ = 0.121
θ = 6.964°

the plane must travel 6.964° north of east to maintain a direct east route...


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Answer 2

This is vector addition. You will have two vectors, which when added, have a resultant that is due east. The net vertical (north/south) component is zero. We don't care what the horizontal (east/west) component is.

Let
θ = angle adjustment south of east that the pilot has to make
N = vertical component

N = 70(sin 60°) + 500(sin θ) = 0
sin θ = -70(sin 60°)/500 = -70(√3/2)/500
sin θ = -70√3 / 1000 = -7√3 / 100

θ = arcsin[-7√3 / 100] ≈ -6.964°

So the pilot needs to set a heading 6.964° South of East to maintain a heading directly east.

Answer 3

Using the Law of Sines:

adjustment = arcsin(500 sin(60) / 70) = approx. 7 degrees (south of east)