My Vector Enigma: Solving Analytically w/ Trig Functions

[Noesis]

My question follows off of the picture attached.

I have been able to answer both questions, but I still have questions of my own.

I had to solve them geometrically using the Law of Cosines and the Law of Sines...why is it that I am not able to break it into components using the trig functions?

Surely it has something to do with the axes not being perpendicular..but what exactly?

As far as the u axis goes...it's just a direction, independent of v, so I don't understand why I cannot solve it using trig functions multiplied by the magnitude.

The answer for the second problem, is about 205 N via the Law of Sines:

Fx = (300*sin(40))/sin(110)

Why can't I simply do cos(30)*300 ?

And when I try to implement a normal perpendicular system, I still get the wrong components. What am I doing wrong?

And how would you solve this analytically with trig functions and not geometrically?

Thank you so much guys.

 

[Noesis]

Also if I shift the bottom axis over 20 degrees to make it perpendicular, and try cos(25)*500 to get the y component, and cos(30)*300 to get the x, then use both to get the magnitude, I actually get a different magnitude than I would have using the Law of Cosines.

It's a bit less.

Not understanding why either.