Writing a parametrization of the cissoid by using $\theta$

Question

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.

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

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

Answer 1

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 $$x=t^2,\qquad y=\frac{t^3}{\sqrt{1-t^2}}.$$