For the vector force system in the photo, I am tasked with finding the component of $F = 600 \ lb$ along the $u$ direction. I originally tried to solve the problem by $$F_u = F \cos{\theta}$$ but when looking at the solution, I see that the law of sines was used such that $$\frac{F_u}{\sin{120}}=\frac{600}{\sin{30}}.$$ My question is, why can't you just find the component of $F$ in the $u$ direction by taking the cosine of the angle between them, like a dot product? The answer in the textbook was $F_u=1039 \ lb$. Interestingly, in this case the component has a greater magnitude than the resultant.
Thanks for the help.