Vector components

Every vector in a plane is the sum of two perpendicular pieces, one along each axis you pick. EK 1.5.A.1 frames it as modeling the vector as the resultant of two perpendicular components, and EK 1.5.A.3 says trigonometry is how you get them.

For a vector of magnitude $A$ at an angle $\theta$ measured from the positive x axis, $A_x = A\cos\theta$ and $A_y = A\sin\theta$. Rebuilding it goes the other way, $A = \sqrt{A_x^2 + A_y^2}$ with $\theta = \tan^{-1}(A_y/A_x)$. A 50 N force at 30 degrees above the horizontal has $A_x = 50\cos 30 = 43.3$ N and $A_y = 50\sin 30 = 25.0$ N.

The angle decides which function goes with which axis. Cosine belongs to the axis the angle is measured from, so measuring that same 30 degrees from the vertical swaps the two formulas. Memorizing that x takes cosine is how people lose the mark. On a ramp of angle $\theta$ the weight splits into $mg\sin\theta$ along the slope and $mg\cos\theta$ into the surface, which looks backwards until you notice the ramp angle is measured from the horizontal while the weight hangs vertically.

The axes are yours to choose, and EK 2.2.B.4 notes that lining one up with the acceleration simplifies the algebra. The inclined plane guide works that case; projectile motion works the launch-angle case.