Why do VR motors like sinusoidal current?
Because a sinusoidal current will make the torque applied to the motor constant, whetever the position of the shaft.

Let's say we have winding A and B, and supply current I. If you apply a square current to both windings, you'll have:


a phase where a current I is applied to winding A, and no current in winding B
a phase where a current I is applied for both A and B, resulting in more power being applied to the motor then in the previous phase
a phase where a current I is applied to winding B, and no current to winding A


So you see, the power given to the motor varies (sometimes I, sometimes 2*I), depending on the position of the shaft, so it will not run smoothly.

To solve this, you have to apply sinusoidal currents. Then, even when both winding are powered, the torque is kept constant.

Why a sine and not a triangle wave ?

Think of what you need to provide when the shaft is at 45° (both windings powered): the power provided to the motor is twice V*I (once for each winding). V and I are linked by Ohm's law (if we assume steady state and rule out inductance). So the power is twice RI². Whereas when the shaft is at 0°, the power is only once RI². Now, how to make this constant ? By setting the I at 45° to sqrt(0.5)*I at 0°. Indeed : 2*R*(sqrt(0.5)I)² = RI². Now, what is sqrt(0.5) ? It is cos(45°). So you can start visualising the sine wave. Basically, what is required is that I² in winding A + I² in winding B is constant. That is the definition of sin and cos.

And yes, basically, it comes from the fact that the motor turns in circle.