How do I convert the polar equation ##r=10 sin theta## to its Cartesian equivalent?

Cartesian coordinates, also known as Rectangular coordinates, are defined in terms of ##x## and ##y##. So, for this problem ##theta## has to be eliminated/converted using basic foundations described by the unit circle and right triangle trigonometry.

##r=10sin(theta)##

Remember that ...

##x=r*cos(theta)##

##y=r*sin(theta)##

##r^2=x^2+y^2##

Multiply both sides of the equation by ##r##

##r*r=10r*sin(theta)##

##r^2=10r*sin(theta)##

##x^2+y^2=10r*sin(theta)##

Use the fact that ##y=r*sin(theta)## to make a substitution.

##x^2+y^2=10y##

##x^2+y^2-10y=0##

The above equation is the Cartesian/Rectangular coordinate equivalent to the given Polar equation.