A jet is flying through a wind that is blowing with a speed

A jet is flying through a wind that is blowing with a speed of 50 mph in the direction of N30 E (see figure). The jet has a speed of 775 mph relative to the air, and the pilot heads the jet in the direction N45 E. Find the true speed of the jet. (Round to the nearest integer 30 450

Solution

Note that the jet heads in a direction 45° west of north, so the jet will be traveling at an angle of 90° + 45° = 135° with respect to the positive x-axis. Thus, the vector that represents the velocity of the jet is:
<775cos(135°), 775sin(135°)>.

The wind vector is 30° east of north, so this vector is just:
<50cos(60°), 50sin(60°)>.

Adding these two vectors together gives:
<775cos(135°) + 50cos(60°), 775sin(135°) + 50sin(60°)>
= <-523, 591.3>.

This vector does have the required magnitude and direction. Maybe you calculated them wrong? The magnitude is:
|<-523, 591.3>| = [523^2 + (591.3)^2] 789.4 mph.

To find the direction, note that <-523, 591.3> is in Quadrant II. Using right triangles, the angle that this vector makes with the negative x-axis is:
tan = 591.3/523 ==> = 48.5°,

which is equivalent to N(90 - 48.5)°W = N41.5°W.

I hope this helps!