Convert the rectangular equation x^2 +(y+2)^2 =4 to a polar equation

Question

Please help me with a step by step so I can find my mistake thank you

Answer 1

In rectangular coordinates, this is a circle of radius = 2 centered at (0,-2).

x = rcos(θ) , y = rsin(θ)

Substituting

r²cos²(θ) + (rsin(θ) + 2)² = 4

r²cos²(θ) + r²sin²(θ) + 4rsin(θ) + 4 = 4

r²(cos²(θ) + sin²(θ)) + 4rsin(θ) = 0

r² + 4rsin(θ) = 0

r = -4sin(θ)

Which is also a circle centered at (0,-2) of radius 2.