Show that simple harmonic motion may be regarded as the projection of uniform circular motion along a diameter of the circle.

(N/A) Consider a particle moving in a circle of radius $R$ with a constant angular speed $\omega$ in a horizontal plane.
The position of the particle at any time $t$ can be represented by the angle $\theta = \omega t$ with respect to a reference diameter.
If we project the position of the particle onto a diameter of the circle,the displacement $y$ of the projection from the center of the circle at time $t$ is given by $y = R \sin(\omega t)$.
This equation $y = R \sin(\omega t)$ represents the displacement of a particle executing simple harmonic motion $(SHM)$.
As shown in the figure,as the particle moves along the circular path through points $A, B, C, D, E, F, G$,its projection on the vertical diameter moves back and forth between the extreme points $S$ and $Q$ through the center $O$. This oscillatory motion of the projection is simple harmonic motion.