Question

An airplane with an air speed of 165 mph is headed 210°. If the plane encounters a 15 mph wind from the west, find the ground speed and course of the plane?

Answer 1

Since a calculator is involved in finding the answer, it makes sense to me to use a calculator capable of adding vectors.

The airplane's ground speed is 158 mph, and its heading is 205.3°.

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A diagram can be helpful. You have enough information to determine two sides of a triangle and the angle between them. This makes using the Law of Cosines feasible for determining the resultant (r) of adding the two vectors.
.. r^2 = 165^2 +15^2 -2*165*15*cos(60°) = 24975
.. r = √24975 ≈ 158.03
Then the angle β between the plane's heading and its actual direction can be found from the Law of Sines
.. β = arcsin(15/158.03*sin(60°)) = 4.7°
Thus the actual direction of the airplane is 210° -4.7° = 205.3°.

The ground speed and course of the plane are 158 mph @ 205.3°.