How do you convert the polar equation #r=3sintheta# into rectangular form?

Question

Carter Anderson · Accepted Answer

#x^2+(y-3/2)^2=9/4#

#x=rcos(theta)#
#y=rsin(theta)#
#r^2=x^2+y^2#

Multiply the equation by #r#

#r*r=3rsin(theta)#

Simplify

#r^2=3rsin(theta)#

Make appropriate substitutions

#x^2+y^2=3y#

Gather all of the terms to the same side

#x^2+y^2-3y=0#

Complete square using the coefficient of #y# variable

#(-3/2)^2=9/4#

Add #9/4# to both sides the equation to keep it balanced. The constant #9/4# allows you make a perfect square trinomial.

#x^2+y^2-3y+9/4=9/4#

Rewrite

#x^2+(y-3/2)^2=9/4#

Check out a tutorial on converting an equation from polar to rectangular

Check out a tutorial on completing the square graphically

Check out a tutorial on completing the square analytically

Nolan Alexander · Answer

undefined To convert the polar equation $ r = 3\sin(\theta) $ into rectangular form, you can use the relationships between polar and rectangular coordinates:

$$ x = r\cos(\theta) $$
$$ y = r\sin(\theta) $$

Substitute $ r = 3\sin(\theta) $ into these equations:

$$ x = 3\sin(\theta)\cos(\theta) $$
$$ y = 3\sin^2(\theta) $$

Using the trigonometric identity $ \sin(2\theta) = 2\sin(\theta)\cos(\theta) $, you can rewrite $ x $ in terms of $ \sin(2\theta) $:

$$ x = \frac{3}{2}\sin(2\theta) $$

Thus, the rectangular form of the polar equation $ r = 3\sin(\theta) $ is $ x = \frac{3}{2}\sin(2\theta) $ and $ y = 3\sin^2(\theta) $.