Expressing $ r = \cot(\theta) $ as an equation in terms of Cartesian coordinates $ (x,y) $.

Question

I need to show this equation

$r = \cot(\theta)$ as $x$,$y$ using the following laws:

$x=r\cos(\theta)$, $y=r\sin(\theta)$

$r^2=x^2+y^2$, $\tan(\theta)=\frac{y}{x}$

This is what I've done :

$$r = \cot(\theta) \\ r = \frac{\cos(\theta)}{\sin(\theta)} \\ r^2=\frac{r\cos(\theta)}{\sin(\theta)}\\ x^2+y^2=\frac{x}{\sin(\theta)}$$

Now, I'm stuck what should I do with $\sin(\theta)$?

Any ideas?

Thanks!

Answer 1

You can also use

$$r=\sqrt{x^2+y^2}$$

and $$\cot{\theta}=\frac{x}{y}$$

instead.

Answer 2

Surely it is just $x^2 + y^2 = (\frac{x}{y})^2$?

By the way, I think you might mean $x^2+y^2$ instead of $x^2=y^2$ in your fourth line down :)

Answer 3

$$ r \tan \theta =1 $$

$$ r^2 \tan^2 \theta =1 $$

$$ ( x^2 +y^2) \cdot \dfrac{y^2}{x^2} = 1 $$

Littus?