Vector magnitude and direction

[magnifik]

Homework Statement

Find the magnitude of the resultant force and the angle it makes with the positive x-axis

draw one vector from origin (0,0) in quadrant one that is 45 degrees (this is 20 lbs)

draw another vector from the origin (0,0) in quadrant four that is 30 degrees (this is 16 lbs)

The Attempt at a Solution

here's what i did initially
20cos(45)i + 20sin(45)j = 10*sqrt(2)i + 10*sqrt(2)j
16cos(-30)i + 16sin(-30)j = 8*sqrt(3)i - 8j
F = (10*sqrt(2)i + 10*sqrt(2)j) + (8*sqrt(3)i - 8j)
|F| = sqrt((10*sqrt(2)i + 10*sqrt(2)j)^2) + (8*sqrt(3)i - 8j)^2)) = 28.66 N
$$\vartheta$$ = (10*sqrt(2) - 8)/(10*sqrt(2) + 8*sqrt(3)) = 12.37 deg

but if you use cosine rule or sine rule, you get different answers...
The angle is 45* + 30* = 75*

Now we use the Cosine rule to find the magnitude of the resultant vector. The resultant vector is 22.1 lbs.

Use the Sin rule and the angle is 90.7* (basically 90*)

which one is right?

 

[coreyB]

The angle of the resultant vector should be between the angles of your initial vectors. Your answer of 12.37 degrees seems reasonable...however, 75 degrees and 90.7 degrees do not.