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?
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
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

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