rewrite the polar equation ( r = 4sin(\theta) ) as a cartesian equation.

Explanation:

Step1: Multiply both sides by ( r )

$$r^2 = 4r\sin(\theta)$$

Step2: Use the conversion formulas ( x = r\cos(\theta) ), ( y = r\sin(\theta) ), and ( r^2=x^2 + y^2 )

Substitute ( r^2=x^2 + y^2 ) and ( y = r\sin(\theta) ) into the equation: $$x^2 + y^2=4y$$

Step3: Rearrange the equation to the standard form of a circle

$$x^2 + y^2-4y = 0$$ Complete the square for the ( y ) - terms. For ( y^2-4y ), we have ( y^2-4y=(y - 2)^2-4 ). So the equation becomes ( x^2+(y - 2)^2-4 = 0 ), or ( x^2+(y - 2)^2=4 )

Answer:

(x^{2}+(y - 2)^{2}=4)