Question

Find a cartesian equation for the curve. r = 9 sin(θ)

Answer 1

For this case we have the following equation:
 r = 9 sin (θ)
 In addition, we have the following change of variables:
 y = r * sine (θ)
 Rewriting the equation we have:
 r = 9 sin (θ)
 r = 9 (y / r)
 r ^ 2 = 9y
 On the other hand:
 r ^ 2 = x ^ 2 + y ^ 2
 Substituting values:
 x ^ 2 + y ^ 2 = 9y
 Rewriting:
 x ^ 2 + y ^ 2 - 9y = 0
 Completing squares:
 x ^ 2 + y ^ 2 - 9y + (-9/2) ^ 2 = (-9/2) ^ 2
 Rewriting:
 x ^ 2 + 1/4 (2y-9) ^ 2 = 81/4
 4x ^ 2 + (2y-9) ^ 2 = 81
 Answer:
 The Cartesian equation is:
 4x ^ 2 + (2y-9) ^ 2 = 81