Answer
$ r(t)=\langle \pm\frac{\sqrt2}{2}, \pm\frac{\sqrt2}{2}, 0\rangle $, $ r(t)=\langle 0, \pm 1, 0\rangle $
Work Step by Step
Solve for where $ z(t)=0$ for $0 \le t < 2\pi $:
$ sintcos2t=0$
$ sint=0$ or $ cos2t=0$
$ t=0, \pi $ or $2t=\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2}$
$ t=0, \pi $ or $ t=\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$
Plug values of $ t $ into $ r(t)$:
$ r(0)=\langle sin(0), cos(0), sin(0)cos(2*0)\rangle $
$ r(0)=\langle 0, 1, 0\rangle $
$ r(\pi)=\langle sin(\pi), cos(\pi), sin(\pi)cos(2*\pi)\rangle $
$ r(\pi)=\langle 0, -1, 0\rangle $
$ r(\frac{\pi}{4})=\langle sin(\frac{\pi}{4}), cos(\frac{\pi}{4}), sin(\frac{\pi}{4})cos(2*\frac{\pi}{4})\rangle $
$ r(\frac{\pi}{4})=\langle \frac{\sqrt2}{2}, \frac{\sqrt2}{2}, 0\rangle $
$ r(\frac{3\pi}{4})=\langle sin(\frac{3\pi}{4}), cos(\frac{3\pi}{4}), sin(\frac{3\pi}{4})cos(2*\frac{3\pi}{4})\rangle $
$ r(\frac{3\pi}{4})=\langle \frac{\sqrt2}{2}, -\frac{\sqrt2}{2}, 0\rangle $
$ r(\frac{5\pi}{4})=\langle sin(\frac{5\pi}{4}), cos(\frac{5\pi}{4}), sin(\frac{5\pi}{4})cos(2*\frac{5\pi}{4})\rangle $
$ r(\frac{5\pi}{4})=\langle -\frac{\sqrt2}{2}, -\frac{\sqrt2}{2}, 0\rangle $
$ r(\frac{7\pi}{4})=\langle sin(\frac{7\pi}{4}), cos(\frac{7\pi}{4}), sin(\frac{7\pi}{4})cos(2*\frac{7\pi}{4})\rangle $
$ r(\frac{7\pi}{4})=\langle -\frac{\sqrt2}{2}, \frac{\sqrt2}{2}, 0\rangle $
