Trig Question (check my answer)?

Question

A man drivers 10 mi E, then 13 mi in a direction 22.6 degrees W of S, then 4 mi at 45 degrees N of W. What is his overall displacement.

I got 14.87 degrees for 8.45 miles West of North as my answer, can anyone confirm this?

Thanks!

Answer 1

10 miles 90 deg: X = 10 * (sin 90), Y = 10 * (cos 90)
13 miles 202.6 deg: X = 13 * (sin 202.6), Y = 13 * (cos 202.6)
4 miles 315 deg: X = 4 * (sin 315), Y = 4 * (cos 315)

Add X components:
10(sin90) - 13(sin22.6) - 4(sin45) = 2.1757
Add Y components:
10(cos90) - 13(cos22.6) + 4(cos45) = -9.1733

Distance:
= SQRT((X*X)+(Y*Y))
= SQRT (2.1757^2+9.1733^2)
= 9.4278

Angle:
= we have positive X and negative Y, therefore angle will be east of south.
= 180 + arctan (X/Y)
= 180 + arctan (2.1757/-9.1733)
= 180 -13.34
= 166.6573 degrees (which is the same as 13.3427 degrees east of south)

Answer 2

X displacement with E = positive =
10 + 13(cos(-(90+22.6)) + 4(cos(90+45)) =
10 + 13(cos(-112.6)) + 4(cos(135)) =
10 + -5 + -2.8284 = 2.1716 miles E

Y displacement with N = positive =
0 + 13(sin(-112.6)) + 4(sin(135)) = (note same angles used)
0 + -12 + 2.8284 = 9.1716 miles South

arctan (2.1716/9.1716) = 13.32 degress east of south
distance = sqrt( sum of squares ) = 9.425 miles

Answer 3

I got 9.43 miles at 13.5 degrees west of south. The sum of my x components was 2.17 miles, and the sum of my y components was -9.17 miles. You might want to double check my angle; I did this without a proper calculator which does trig.