Answer
(a) $x = sin~t$
$y = csc~t$
for $t$ in $(0,\pi)$
We can see the graph below.
(b) $y = \frac{1}{x}$
for $x$ in $(0,1]$
Work Step by Step
(a) $x = sin~t$
$y = csc~t$
When $t = \frac{\pi}{10}$:
$x = sin~\frac{\pi}{10} = 0.31$
$y = csc~\frac{\pi}{10} = 3.24$
When $t = \frac{\pi}{6}$:
$x = sin~\frac{\pi}{6} = \frac{1}{2}$
$y = csc~\frac{\pi}{6} = 2$
When $t = \frac{\pi}{4}$:
$x = sin~\frac{\pi}{4} = \frac{\sqrt{2}}{2}$
$y = csc~\frac{\pi}{4} = \sqrt{2}$
When $t = \frac{\pi}{3}$:
$x = sin~\frac{\pi}{3} = \frac{\sqrt{3}}{2}$
$y = csc~\frac{\pi}{3} = \frac{2\sqrt{3}}{3}$
When $t = \frac{\pi}{2}$:
$x = sin~\frac{\pi}{2} = 1$
$y = csc~\frac{\pi}{2} = 1$
When $t = \frac{2\pi}{3}$:
$x = sin~\frac{2\pi}{3} = \frac{\sqrt{3}}{2}$
$y = csc~\frac{2\pi}{3} = \frac{2\sqrt{3}}{3}$
We can see the graph below.
Note that $x=0$ is an asymptote.
(b) $x = sin~t$
$y = csc~t = \frac{1}{sin~t}$
Therefore: $~~y = \frac{1}{x}$
Since $t$ in $(0,\pi)$, then $x$ in $(0,1]$
