Answer: r=2sinθ.
Step-by-step explanation:To convert to polar coordinates, we use the relationships
xy=rcosθ=rsinθ.
Substituting these into the given equation, we find
x2+y2(rcosθ)2+(rsinθ)2r2cos2θ+r2sin2θ=2y=2(rsinθ)=2rsinθ
Factoring and applying the Pythagorean identity, we obtain
r2(cos2θ+sin2θ)r2(1)r2=2rsinθ=2rsinθ=2rsinθ
Dividing both sides by r yields
r2rr=2rsinθr=2sinθ
Thus, a polar equation representing the given function is r=2sinθ.