Graph $r=\cot(\theta)\csc(\theta)$.

Question

So I am stuck on a graphing question in polar coordinates. I am not sure how to graph the equation even after writing down a few points:

$\cot(0)\csc(0) $DNE

$\cot(\dfrac {\pi}{4})\csc(\dfrac {\pi}{4})= \sqrt{2}$,

$\cot(\dfrac {\pi}{2})\csc(\dfrac {\pi}{2})= 0,$

$\cot(\dfrac {3\pi}{4})\csc(\dfrac {3\pi}{4})= -\sqrt{2}$,

$\cot(\pi)\csc(\pi) $DNE,

and so on.

Obviously I can just graph it online, but can someone explain the mechanics to how to graph it without using a graphing calculator or desmos? Thanks!

Answer 1

It might help to convert the equation to rectangular form using the identities

1. $x=r\cos\theta$
2. $y=r\sin\theta$

\begin{eqnarray} r&=&\cot\theta\csc\theta\\ r&=&\frac{\cos\theta}{\sin^2\theta}\\ r\sin^2\theta&=&\cos\theta\\ r^2\sin^2\theta&=&r\cos\theta\\ y^2&=&x \end{eqnarray}

ADDENDUM: Note that multiplying both sides by $r$ introduces an extraneous solution at the origin. ADDENDUM-2: As Oscar Lanzi points out, there is a value of $\theta$ which includes the origin, so the origin is NOT an extraneous solution.

Answer 2

A good practice is to render the trig functions in terms of sines and cosines, thus

$r=\frac{1}{\sin\theta}\frac{\cos\theta}{\sin\theta}$

Then, remembering that $x=r\cos\theta, y=r\cos\theta$, put in:

$r\sin\theta=\frac{\cos\theta}{\sin\theta}=\frac{r\cos\theta}{r\sin\theta}$

and continue from there. Note that unlike what is suggested in the other answer, the origin is a good solution given the proper value of $\theta$. Can you find this value of $\theta$?