smooth a motor movement
What you're looking for is called a Trapezoidal velocity profile.



There is no need to use sin waves. Smooth motion can be achieved by simply linearly ramping up the velocity to its maximum, holding it there, then letting it ramp down again. Many CNC milling machines and robots use this type of profile.

If you want super smooth motion, you can go for an S-Curve profile.



The maths for this is considerably harder, but the motion is beautiful. In this video they show how the two motion profiles affect a glass of water.