Converting polar curve to to cartesian

Question

r= (1)/(2-sinθ)

Answer 1

we desire to have an equation involving only variables x and y.

in general, sinθ = y/r

r= (1)/(2-sinθ)

r = √(x²+y²) = 1 / (2 - y/r)

 

r               =  r / (2r-y)

1               = 1/(2r-y)

2r - y = 1

y = 2r - 1

y = 2√(x²+y²) - 1

and this can be manipulated further according to taste.

Answer 2

 

x = r cosθ = cosθ/(2 - sinθ) 

 

y = r sinθ = sinθ/(2 - sinθ)