Ok, the obvious parametrization using the parameter $\theta$ is $$\gamma(\theta) = (\cos \theta,\sin \theta)\qquad\theta\in \left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}2,2\pi\right]$$
after drawing some circles on a paper I observed that $\theta$ is twice bigger than $t$ but I'm not sure about it can you please tell me how can I show that $\theta$ = $2t$ if $\theta$ happens to be indeed $2t$?
It is called the inscribed angle theorem.
If you have two points on a circle $A$ and $B$, and you put a third one $C$, then the angle $\widehat{ACB}$ will be half the angle $\widehat {AOB}$ where $O$ is the center of the circle.
So in your case, $$\theta=2t.$$