(a)

$\begin{array}{l}\mathrm{x}=-2\sin (3\mathrm{t}+\frac{\mathrm{\pi }}{3})=+2\cos (3\mathrm{t}+\frac{\mathrm{\pi }}{3}+\frac{\mathrm{\pi }}{2})\\ =2\cos (3\mathrm{t}+\frac{5\mathrm{\pi }}{6})\end{array}$
$\mathrm{Comparing} \mathrm{it} \mathrm{with} \mathrm{the} \mathrm{standard} \mathrm{SHM} \mathrm{equation}:$
$\mathrm{x}=\mathrm{Acos}\left(\frac{2\mathrm{\pi }}{\mathrm{T}}\mathrm{t}+\mathrm{\varphi }\right), \mathrm{then} \mathrm{we} \mathrm{get}:$
$\mathrm{Amplitude}, \mathrm{A}=2 \mathrm{cm}$
$\mathrm{Phase} \mathrm{angle}, \mathrm{\varphi }=\frac{5\mathrm{\pi }}{6}=150°$
$\mathrm{Angular} \mathrm{velocity}, \mathrm{\omega }=\frac{2\mathrm{\pi }}{\mathrm{T}}=3 \mathrm{rad}/\sec$

The motion of the particle is plotted as shown in the following figure.

(b)

$\mathrm{x}=\cos \left(\frac{\mathrm{\pi }}{6}-\mathrm{t}\right)=\cos \left(\mathrm{t}-\frac{\mathrm{\pi }}{6}\right)$
$\mathrm{Comparing} \mathrm{it} \mathrm{with} \mathrm{the} \mathrm{standard} \mathrm{SHM} \mathrm{equation}:$
$\mathrm{x}=\mathrm{A} \cos \left(\frac{2\mathrm{\pi }}{\mathrm{T}}\mathrm{t}+\mathrm{\varphi }\right), \mathrm{then} \mathrm{we} \mathrm{get}:$
$\mathrm{Amplitude}, \mathrm{A}=1 \mathrm{cm}$
$\mathrm{Phase} \mathrm{angle}, \mathrm{\varphi }=-\frac{\mathrm{\pi }}{6}=-30°$
$\mathrm{Angular} \mathrm{velocity}, \mathrm{\omega }=\frac{2\mathrm{\pi }}{\mathrm{T}}=1 \mathrm{rad}/\mathrm{s}$

The motion of the particle is plotted as shown in the following figure.

(c)

$\begin{array}{l}\mathrm{x}=3\sin (2\mathrm{\pi t}+\frac{\mathrm{\pi }}{4})\\ =-3\cos [(2\mathrm{\pi t}+\frac{\mathrm{\pi }}{4})+\frac{\mathrm{\pi }}{2}]=-3\cos (2\mathrm{\pi t}+\frac{3\mathrm{\pi }}{4})\end{array}$

Comparing it with the standard SHM equation: $\mathrm{x}=\mathrm{A} \cos \left(\frac{2\mathrm{\pi }}{\mathrm{T}}\mathrm{t}+\mathrm{\varphi }\right)$

Then we get:
Amplitude, A = 3 cm
$\mathrm{Phase} \mathrm{angle}, \mathrm{\varphi }=\frac{3\mathrm{\pi }}{4}=135°$
$\mathrm{Angular} \mathrm{velocity}, \mathrm{\omega }=\frac{2\mathrm{\pi }}{\mathrm{T}}=2\mathrm{\pi } \mathrm{rad}/\mathrm{s}$

The motion of the particle is plotted as shown in the following figure.

(d)
x = 2cos πt

Comparing it with the standard SHM equation: $\mathrm{x}=\mathrm{A} \cos \left(\frac{2\mathrm{\pi }}{\mathrm{T}}\mathrm{t}+\mathrm{\varphi }\right)$,

then we get:
Amplitude, A = 2 cm
Phase angle, Φ = 0
Angular velocity, ω = π rad/s
The motion of the particle is plotted as shown in the following figure.