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 \mathrm{t} is in \mathrm{s}).
(a) x=-2 \sin (3 t+\pi / 3)
(b) x=\cos (\pi / 6-t)
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 \mathrm{t} is in \mathrm{s}).
(a) x=-2 \sin (3 t+\pi / 3)
(b) x=\cos (\pi / 6-t)

(a) x=-2 \sin (3 t+\pi / 3) =2 \cos (3 t+\pi / 3+\pi / 2) =2 \cos (3 t+5 \pi / 6)

On comparing the above equation with the standard equation, x=A \cos (\omega t+\Phi),

Amplitude will be A=2 \mathrm{~cm} (radius of the circle)

Angular velocity will be \omega=3 \mathrm{rad} / \mathrm{s}

Phase angle will be \Phi=5 \pi / 6=150^{\circ}

(b) x=\cos (\pi / 6-t) =\cos (t-\pi / 6)

On Comparing this equation with A \cos (\omega t+\Phi), we get,

Phase angle as \Phi=-\pi / 6=-30^{\circ}

Amplitude as A=1 \mathrm{~cm} (radius of the circle)

Angular velocity as \omega=1 \mathrm{rad} / \mathrm{s}