find a polar equation for the curve represented by the given cartesian equation. (assume 0 ≤ θ < 2π.)\n\nx²…

Explanation:

Step1: Recall polar - Cartesian conversion formulas

We know that (x = r\cos\theta) and (y = r\sin\theta), and (x^{2}+y^{2}=r^{2}).

Step2: Substitute into the given Cartesian equation

Given (x^{2}+y^{2}=8y). Substitute (x = r\cos\theta), (y = r\sin\theta) and (x^{2}+y^{2}=r^{2}) into the equation. We get (r^{2}=8r\sin\theta).

Step3: Simplify the equation

Since (r\neq0) (if (r = 0), it represents the origin which is also included in the non - zero solutions of the simplified equation). Divide both sides of the equation (r^{2}=8r\sin\theta) by (r) (for (r\neq0)). We obtain (r = 8\sin\theta).

Answer:

(r = 8\sin\theta)