So, I'm trying to understand this.
In the following we've an image, with a vector $v_0$ and we can see the components of the vectors have trigonometric ratios such as $\sin\theta$ and $\cos\theta$ for vertical and horizontal components respectively.
But how can we proof, that the vertical component be $v_0\sin\theta$ and horizontal component be $v_0\cos\theta$?
I know that when we add two vectors, the magnitude of the resultant vector is $$=\sqrt {a^2+b^2+2ab\cos \theta}$$ through the triangle law, but how can we prove this?
These results are coming from the definition of sine and cosine functions.
Your hypotenuse is the norm of your vector and the components are the adjacent and opposite sides of a right triangle.
Thus the components are the norm multiplied by sine and cosine of the angle between your vector and the positive direction of the $x$-axis.