Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial $(t = 0)$ position of the particle,the radius of the circle,and the angular speed of the rotating particle. For simplicity,the sense of rotation may be fixed to be anticlockwise in every case: ($x$ is in $cm$ and $t$ is in $s$).
$(a)\; x = -2 \sin (3t + \pi/3)$
$(b)\; x = \cos (\pi/6 - t)$
$(c)\; x = 3 \sin (2\pi t + \pi/4)$
$(d)\; x = 2 \cos \pi t$

(N/A) The general equation for $SHM$ is $x = A \cos (\omega t + \phi)$.
$(a)\; x = -2 \sin (3t + \pi/3) = 2 \cos (3t + \pi/3 + \pi/2) = 2 \cos (3t + 5\pi/6)$.
Comparing with $x = A \cos (\omega t + \phi)$,we get $A = 2 \text{ cm}$,$\omega = 3 \text{ rad/s}$,and $\phi = 5\pi/6 = 150^{\circ}$.
$(b)\; x = \cos (\pi/6 - t) = \cos (t - \pi/6)$.
Comparing with $x = A \cos (\omega t + \phi)$,we get $A = 1 \text{ cm}$,$\omega = 1 \text{ rad/s}$,and $\phi = -\pi/6 = -30^{\circ}$.
$(c)\; x = 3 \sin (2\pi t + \pi/4) = 3 \cos (2\pi t + \pi/4 - \pi/2) = 3 \cos (2\pi t - \pi/4)$.
Comparing with $x = A \cos (\omega t + \phi)$,we get $A = 3 \text{ cm}$,$\omega = 2\pi \text{ rad/s}$,and $\phi = -\pi/4 = -45^{\circ}$.
$(d)\; x = 2 \cos (\pi t)$.
Comparing with $x = A \cos (\omega t + \phi)$,we get $A = 2 \text{ cm}$,$\omega = \pi \text{ rad/s}$,and $\phi = 0$.