For projectile motion, horizontal velocity = #v_0Cos(theta)#. The reason we use Cos is because x forms the base of the triangle and is therefore the adjacent side. #Cos(theta) = {adj}/{hyp}#
Then, from Newton's first law, #v_{0x} = v_{fx}# in the absence of friction or air resistance.
Key understanding, horizontal acceleration #a_x# = 0. There is no horizontal acceleration in projectile motion, only horizontal velocity.
For the y components, #v_{0y}=v_0Sin(theta)#
#v_{fy} = 0# because the projectile will come to a momentary stop at the top of its parabolic path.
Another key understanding: the only acceleration in the system is in the downward y direction and that's due to gravity. #a_y=-9.8m/s^2#. That acceleration points downward as a vector all the way along the parabola.
