Speed = 283 miles per hour
Direction = 4.87 degrees south of west.
This is an example of vector addition. Starting at the origin you plot the 1st vector (or do the math to determine the endpoint of the vector), then for the next vector you perform the same thing except you start where the previous vector ended. After you've handled all the vectors, the result is a single vector from the origin to the end point of the last vector. So let's do this.
Airplane is traveling due west at 258 mph.
The vector is from (0,0) to (-258,0).
; If you really want the math, you add (258*cos(180), 258*sin(180)) = (258*(-1), 258*0) = (-258, 0) to the starting point of (0,0).
Wind is blowing towards 45 degrees south of west at 34 mph.
The direction is 180+45 = 225 degrees. So we need to add (34*cos(225), 34*sin(225)) = (34*(-0.707106781), 34*(-0.707106781)) = (-24.04163056, -24.04163056) to the endpoint from the previous step which was (-258, 0), giving (-282.0416306, -24.04163056)
So we now have a right triangle with the legs being of lengths -282.0416306 and -24.04163056. The effective speed can be calculated using the Pythagorean theorem.
S = sqrt(-282.0416306^2 + -24.04163056^2)
S = sqrt(79547.4813691368 + 578)
S = sqrt(80125.4813691368)
S = 283.0644474
The direction will be
D = arctan(-24.04163056/-282.0416306) + 180
D = arctan(0.085241425) + 180
D = 4.872196061 + 180
D = 184.872196061
So the plane's direction is 4.87 degrees south of west.
