Converting the equation of a Cartesian curve to Polar Form

Polar coordinates use a radius and an angle to locate a point in a plane, as opposed to Cartesian coordinates which use an x-value and a y-value.
The radius, often denoted as ‘r’, represents the distance from the origin (0,0) to the point, and the angle, usually noted as ‘θ’, measures the rotation needed to get from the positive x-axis to the line connecting the point and the origin.

The relationships between Cartesian coordinates and polar coordinates are often written as x = rcos(θ) and y = rsin(θ).
To convert a set of Cartesian coordinates (x, y) to polar coordinates (r, θ), the radius r can be found using the formula r = √(x² + y²) and the angle θ can be found by using the atan2(y, x) function if available or arctan(y/x) if x>0 and the correct quadrant correction if not.
Remember to account for the sign of x and y while calculating the angle θ to make sure it lands in the correct quadrant.

An equation in Cartesian form can be converted to polar form by substitifying x and y in terms of r and θ using the above formulas.
After substitution, if there are any remaining x or y terms, they can be replaced using the Pythagorean Identity r² = x² + y².
Try to simplify the resulting equation, possibly by multiplying through by r or sin(θ) or cos(θ) to remove denominators, to be in terms of r as much as possible.
For example, the equation for a circle in Cartesian form is x² + y² = r², which is already in polar form!

Test several points to sanity-check your conversion and make sure it makes sense. The same point should hold in both the Cartesian and the polar equation.
Graphing a polar equation can also be insightful as the shape of a polar graph can sometimes help elucidate the logic behind the equation conversion. For instance, the polar equation for a circle centered at the origin is simply r = radius.

Finally, bear in mind that polar coordinates have a close relationship with complex numbers. Writing a complex number in polar form (modulus-argument form) can often simplify arithmetic operations (like multiplication or division) and powers.