Rewrite the polar equation r=3sin(θ) as a Cartesian equation.

Answer 1

r = 3 sin θ

r / 3 = sin θ

r / 3 = y / r

r² = 3y

x² + y² = 3y

x² + y² - 3y = 0

x² + y² - 3y + (3/2)² = (3/2)²

x² + (y - 3/2)² = (3/2)²

Answer 2

Recall that x = rcosθ, y = rsinθ, and r² = x² + y².

So, if r = 3sinθ, multiply both sides by r to get r² = 3rsinθ.

Therefore, x² + y² = 3y

So, x² + y² - 3y = 0.

Completing the square on the y-terms, we have x² + (y² - 3y + 9/4) = 9/4

So, (x - 0)² + (y - 3/2)² = (3/2)². This is a circle with center (0, 3/2) and radius 3/2.