Answer
$ a.\quad$ see image
$ b.\quad \mathrm{r}^{\prime}(t)=(-\sin t) \mathrm{i}+(\cos t)\mathrm{j}$,
$ c.\quad$ see image
Work Step by Step
$a.$
Parametric equations: $\mathrm{r}(t):\quad\left\{\begin{array}{ll}
x=1+\cos t & \\
y=2+\sin t &
\end{array}\right.$
Apply $ \sin^{2}A+\cos^{2}A=1:$
$(x-1)^{2}=\cos^{2}t,$
$(y-2)^{2}=\sin^{2}t\qquad $so
$(x-1)^{2}+(y-2)^{2}=1,$
a circle with radius 1, center at (1,2)
see image
$b.$
Differentiate $(\displaystyle \frac{d}{dt})$ each component function
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{l}
x=-\sin t\\
y=\cos t
\end{array}\right.$
$\mathrm{r}^{\prime}(t)=(-\sin t) \mathrm{i}+(\cos t)\mathrm{j}$,
$c. $
For $t_{o}$ = $\pi/6,$
position vector : black$\quad \mathrm{r}(t_{o})$=$\displaystyle \langle\frac{2+\sqrt{3}}{2}, \frac{5}{2}\rangle$
tangent vector: red$\quad \mathrm{r}^{\prime}(t_{o})$=$\displaystyle \langle-\frac{1}{2},\frac{\sqrt{3}}{2}\rangle$
