Answer
The curve intersects the $z$-axis at $\left( {0,0,t} \right)$ for $t = \pm \pi , \pm 3\pi , \pm 5\pi ,...$.
Work Step by Step
A space curve intersects the $z$-axis if $x=0$ and $y=0$. Thus, for the space curve ${\bf{r}}\left( t \right) = \left( {\sin t,\cos t/2,t} \right)$ to intersect the $z$-axis, we must have
$\sin t = 0$ ${\ \ }$ and ${\ \ }$ $\cos t/2 = 0$
The solutions are $t = \pm \pi , \pm 3\pi , \pm 5\pi ,...$. Thus, the points of intersection are located at $\left( {0,0,t} \right)$ for $t = \pm \pi , \pm 3\pi , \pm 5\pi ,...$.
