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# How do you find an equivalent equation in rectangular coordinates r = 1 + 2 sin x?

Mar 8, 2018

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

#### Explanation:

The relation between polar coordinates $\left(r , \theta\right)$ and rectangular coordinates $\left(x , y\right)$ is

$x = r \cos \theta$ and $y = r \sin \theta$ i.e. ${x}^{2} + {y}^{2} = {r}^{2}$

Hence, we can write $r = 1 + 2 \sin x$

as $\sqrt{{x}^{2} + {y}^{2}} = 1 + \frac{2 y}{\sqrt{{x}^{2} + {y}^{2}}}$

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

The graph appears as follows (drawn using tool from Wolform):