TITLE: Writing a parametrization of the cissoid by using $\theta$
QUESTION [6 upvotes]: The cissoid of Diocles is the curve whose equation in terms of polar coordinates $(r,\theta)$ is
$$r = \sin\theta \tan\theta, −\frac{\pi}{2}<\theta <\frac{\pi}{2}$$
Write down a parametrization of the cissoid using $\theta$ as a parameter and show that
$$\gamma(t)=\left(t^2, \frac{t^3}{\sqrt{1-t^2}}\right)$$ for $-1\lt t\lt 1$ 
is a parametrization of it. 

I researched what is the cissoid of Diocles. And I reached the following graph result; 


I found the question a differential geometry textbook while I am studying by myself . 
Please help me solving the question. Thank you:)

REPLY [6 votes]: You almost have the answer already. Recall the definition of polar coordinates:
$$x=r\cos\theta,\qquad y=r\sin\theta.$$
Now you have an expression for $r$ in terms of $\theta$. Replacing it in these formulas we get
\begin{align}
&x=\sin\theta\tan\theta\cdot\cos\theta=\sin^2\theta,\\
&y=\sin\theta\tan\theta\cdot\sin\theta=\frac{\sin^3\theta}{\cos\theta}.
\end{align}
This is a parameterization of $x$ and $y$ by $\theta$. Now if instead of $\theta$ we use $t=\sin\theta$ as a new parameter, these formulas will transform into
$$x=t^2,\qquad y=\frac{t^3}{\sqrt{1-t^2}}.$$