How to write r=4 sin(theta) with rectangular coordinates?

Question

How can I write the equation r= 4 sin(theta) with rectangular coordinates? I know that I need to convert this polar equation to an equation with x and y, but I do not understand how to do it.

Thank you in advance for your help!

Answer 1

Write the equation r = 4sinθ with rectangular coordinates.

Remember the identities.

x = rcosθ
y = rsinθ
x² + y² = r²
____________

r = 4sinθ

Multiply both sides by r.

r² = 4rsinθ

x² + y² = 4y
x² + y² - 4y = 0
x² + (y² - 4y + 4) = 0 + 4

x² + (y - 2)² = 4

This is the equation of a circle with center (0,2) and radius 2.

Answer 2

r= 4 sin(theta) = 4y/r, since y = rsin(theta)
Multiply both sides by r,
r^2 = 4y
x^2+y^2 = 4y, since r^2 = x^2+y^2

Put in standard form,
x^2+(y-2)^2 = 4

Answer 3

x = 4 Sin(theta) Cos(alpha)
y = 4 Sin(theta) Sin(alpha)