Convert the polar equation to rectangular form.r = 2 sin(𝜃)

Sure, here are the steps to convert the polar equation r = 2sin(θ) to rectangular form:

1. Recall the relationship between polar and rectangular coordinates: x = rcos(θ) and y = rsin(θ).

2. Substitute rsin(θ) for y in the polar equation: r = 2sin(θ) becomes r = 2y/r.

3. Multiply both sides by r to get rid of the fraction: r^2 = 2y.

4. Recall that r^2 = x^2 + y^2 in rectangular coordinates.

5. Substitute x^2 + y^2 for r^2 in the equation from step 3: x^2 + y^2 = 2y.

6. Rearrange the equation to standard form of a circle: x^2 + y^2 - 2y = 0.

7. To complete the square, add the square of half the coefficient of y to both sides: x^2 + (y - 1)^2 = 1.

So, the rectangular form of the polar equation r = 2sin(θ) is x^2 + (y - 1)^2 = 1.