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

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Answer 1 · 2018-03-08T13:38:36

$x^2+y^2=sqrt(x^2+y^2)+2y$

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

$x=rcostheta$ and $y=rsintheta$ i.e. $x^2+y^2=r^2$

Hence, we can write $r=1+2sinx$

as $sqrt(x^2+y^2)=1+(2y)/sqrt(x^2+y^2)$

or $x^2+y^2=sqrt(x^2+y^2)+2y$

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

enter image source here

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