Question

A force of magnitude 15 N is the resultant of two forces, one of which has a magnitude of 8 N and acts at an angle of 30° to the resultant. Find the magnitude and direction of the other force.

Answer 1

You just need to solve a triangle. The other force z
|z|^2 = 8^2 + 15^2 - 2*8*15 cos30° = 81.15
|z| ≈ 9
z makes an angle θ with the resultant, where
8^2 = |z|^2 + 15^2 - 2*|z|*15*cosθ
θ = 26.36°, but points toward the resultant, so θ = -26.36°

Answer 2

or, let the resultant v be aligned with the x-axis. Then
v = (15,0)
and the other force u = (6.93,4.00)
So, you want z=(x,y) such that
(x+6.93) = 15
(y+4.00) = 0
z = (8.07,-4.00)
|z| = 9.01
tanθ = y/x = -4/9
θ = -23.94°
Hmmm. I guess I was a bit off above, eh?
Well, you can double-check my work and see which is correct.