Convert a polar equation into parametric

Question

I am getting hung up on the terminology in Calc 2 and making this subject much harder than it needs to be.

Question: Convert $r=7\sin(\theta)-2$ into parametric.

Does that simply mean putting it into the $x, y$ form?

$x=r\cos(\theta)$ --> $$x=(7\sin(\theta)-2)(\cos(\theta))$$ $y=r\sin(\theta)$ --> $$y=(7\sin(\theta)-2)(\sin(\theta))$$

Answer 1

Consider $r = 7 \sin(\theta)$,

\begin{align*} r^2 &= 7r \sin(\theta)\\ x^2 + y^2 &= 7y\\ x^2 &= 7y - y^2 \end{align*}

Hence $x^2 = t$ and $7y - y^2 = t$. Using quadratic equation, we have $$x = \pm \sqrt{t} \hspace{30pt} \mbox{and} \hspace{30pt} y = \frac{7 \pm \sqrt{49 - 4t}}{2}$$

Answer 2

We can solve this problem with Mathematica:

CoordinateTransform[ 
     "Polar" -> "Cartesian", {7 Sin[θ] - 2, θ}]