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

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