Fundamentals of Polar Co-ordinates

The polar co-ordinate system provides an alternative way of representing points in a plane, often making calculations and geometric reasoning easier.
A point P in the plane has polar co-ordinates (r, θ), where r is the distance of P from the origin (O) and θ is the angle from the positive x-axis to the line connecting O and P.
r>0 is the positive (or outward) direction from the origin, while r<0 is the negative (or inward) direction from the origin.
θ is usually measured in radians but can also be given in degrees.

Conversion between Cartesian and Polar Co-ordinates

To convert from polar co-ordinates (r, θ) to Cartesian co-ordinates (x, y), use the equations: x = r cos(θ), y = r sin(θ).
To convert from Cartesian co-ordinates to polar co-ordinates, use the equations: r = √(x²+y²), θ = tan⁻¹(y/x).
In the conversion from Cartesian to polar, note the quadrant of the origin point to adjust the value of θ.

The equation of a polar curve relates r and θ.
Polar equations often provide a more compact and natural representation of curves like circles, spirals, and cardioids.
Curves fully defined by a function r=F(θ) are typically traced out as θ increases from 0 to 2π, with note of any periodic repetitions or axial symmetries.

Derivatives of polar functions can be obtained, though the process is more complex than in Cartesian coordinates because both r and θ can change.
The area enclosed by a polar curve, r = F(θ) from θ = α to θ = β, can be found using the formula: Area = ½ ∫ [(F(θ))² dθ] from α to β. The factor of ½ accounts for the duplicating effect of radial symmetry when integrating in polar coordinates.
The length of a polar curve from θ = α to θ = β can be expressed as Length = ∫√[r² + (dr/dθ)²]dθ from α to β. The amount by which r and θ change together affects the actual path length.

Complex numbers are often easier to manipulate using polar representation z = r(cos θ + i sin θ) where r is the magnitude and θ is the argument of z.
The polar form of a complex number aids in multiplication and division, as well as raising to power and finding roots.
Euler’s Formula, e^(iθ) = cos θ + i sin θ, provides a particularly convenient bridge between complex numbers and polar co-ordinates.