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
