# How do you convert r^2 theta=1 into cartesian form?

Oct 1, 2016

$y = x \tan \left(\frac{1}{{x}^{2} + {y}^{2}}\right)$, representing a spiral from the origin that turns forever, to reach the origin..

#### Explanation:

As ${r}^{2} > 0$, its reciprocal $\theta > 0$

Using the conversion formula #r(cos theta, sin theta) = (x, y),

$\theta = {\tan}^{- 1} \left(\frac{y}{x}\right) = \frac{1}{{x}^{2} + {y}^{2}} \to y = x \tan \left(\frac{1}{{x}^{2} + {y}^{2}}\right)$.

I want variation of $\theta \in \left(0 , \infty\right)$.

The graph is a spiral that approaches origin only in the limit,

as $x , y \to \infty$.. .

.