How are polar and rectangular planes related?

Polar coordinates (r,ϑ) and Cartesian coordinates (x, y) are related as follows:

Cartesian - > Polar

x = r cos (ϑ)

y = r sin (ϑ)

Polar - > Cartesian

r^2 = x^2 + y^2

ϑ = arctan (y/x)

The polar coo's can be extended to cylindrical by introduction a dimension z - which is perpendicular to the polar plane, it follows that:

z = z

in between the systems (r,ϑ, z) and (x, y, z)

The polar coordinates are particularly useful when a problem has circular or cylindrical symmetry, i. e. it is usually "easier" to perform calculations on a circle (for example) using the polar coordinates.