Transform the following polar equation into an equation in rectangular coordinates: r = -4 sinθ

Solution:

Given r = -4 sinθ -----(1)

We have x = r cos θ; y = r sin θ such that x² + y² = r²

Now multiply both sides of equation (1) by r,

⇒ r² = -4r sinθ

⇒ x² + y² = -4r sinθ

⇒ x² + y² = -4y 

⇒ x² + y² + 4y = 0

Thus the equation in rectangular coordinates for the polar equation r = -4 sinθ is x² + y² + 4y = 0

Transform the following polar equation into an equation in rectangular coordinates: r = -4 sinθ

Summary:

Transform the polar equation r = -4 sinθ into an equation in rectangular coordinates, we get x² + y² + 4y = 0.