An airplane is flying on a bearing of 265° at 420 mph. a wind is blowing with a bearing of 210° at 30 mph. find the actual speed and direction of the plane.

In both cases, y and x components are solved and added. The resulting hypotenuse is then computed and its direction calculated.

Airplane:
Ф = 270 - 265 = 5°
Hypotenuse, h = 420 mph
y component = 420 Sin Ф = 420 Sin 5 = 36.61 mph
x component = 420 Cos Ф = 420 Cos 5 = 418.40 mph

Wind
Ф = 270 -210 = 60°
Hypotenuse, h = 30 mph
y component = 30 Sin 60 = 25.98 mph
x component = 30 Cos 60 = 15 mph

Total y = 36.61+25.98 = 62.59 mph
Total x = 418.40+15 = 433.40 mph

Resulting speed of airplane = Sqrt (y^2+x^2) = Sqrt (62.59^2+433.40^2) = 437.9 mph
Direction = 270 -[tan^-1 (y/x)] = 270 -[tan^-1(62.59/433.40)] = 261.78°