Question

convert the polar equation r=sin(theta) to the cartesian form and hence show that r=sin(theta) is a circle with centre (0;0,5) and radius 0,5

Answer 1

x=rcosθ, y=rsinθ converts polar to Cartesian, x²+y²=r².

r=sinθ, so r²=rsinθ=y, therefore since r²=x²+y²=r², y=x²+y².

This can be written x²+y²-y=0, 

x²+y²-y+¼-¼=0,

x²+(y-½)²=¼, which is the equation of a circle of radius ½ and centre (0,½).