Question

A plane flies 430 miles at a bearing of S27°E, then turns and flies N63°E for 135 miles. Find the distance and the bearing from the starting?

Answer 1

After you make your sketch, and after you enter your angles, you should realize that the plane makes a 90 degree turn.
So you just have a right-angled triangle.
Use Pythagoras to find the distance.
Let the angle opposite the 135 side be θ
then tanθ = 135/430
θ = ...

so your direction is S (θ+27) E

let me know what you got

Answer 2

Your distance of 450.69 miles is correct
However, the angle of 17.43 is the angle within the triangle.
You must add the original direction of 27 degrees to get S 44.3 E