The answer is $\pi \leq t<2 \pi$.
Work Step by Step
The circle is traversed once in the counterclockwise direction as $t$ varies over an interval $[0,2 \pi )$. At $t=\pi$, we have $ c(\pi)=(-1,0)$, so the lower half of the circle starts from $\pi$. Hence, for $\pi \leq t<2 \pi$, the equation $c(t)=(\cos t, \sin t)$ traces the lower half of the unit circle.