I must have made a mistake, can you guys help me find it?

Imagine a turntable, of fixed radius, rotating with constant angular acceleration.

I was trying to plot the motion of an object stuck to the edge of the turntable. So let's consider the displacement-time graph of the object in the x-direction.

We know that the graph of radians-time is quadratic. The displacement-time graph is the "sine" of the radians-time graph. But in doing so, the gradient of the sine graph abruptly changes value.

This implies a sudden change in velocity, and thus infinite acceleration.

But of course, we know that we can angularly accelerate turntables without infinite acceleration.

I haven't been able to resolve this issue.