Due to the periodic nature of angles and the possibility of negative radial distances.

Changing the sign of 'r' reflects the point across the origin (pole).

The arctangent function only gives angles in the 1st and 4th quadrants; adjustment ensures the correct quadrant for θ.

A complex number a + bi is represented by the point (a, b) on the complex plane.

The real and imaginary parts of a complex number can be expressed using polar coordinates: x = r*cos(θ), y = r*sin(θ).

Polar coordinates are useful for circular or rotational symmetry.

Using the wrong units can lead to incorrect conversions and calculations.

Converting between polar and Cartesian coordinates, understanding multiple representations of polar coordinates, complex numbers in polar form, and quadrant awareness.

a + bi

r(cos θ + i sin θ)

Coordinates defined by a distance (r) from the origin and an angle (θ) from the positive x-axis.

The origin (0,0) in the polar coordinate system.

The positive x-axis in the polar coordinate system, used as the reference for measuring angles.

The radial distance from the pole to the point.

The angle measured counterclockwise from the polar axis.

A number of the form a + bi, where 'a' and 'b' are real numbers, and i is the imaginary unit (√-1).

i = √-1

A plane where the x-axis represents the real part of a complex number, and the y-axis represents the imaginary part.

The distance from the origin to the point representing the complex number in the complex plane.

The angle between the positive real axis and the line connecting the origin to the complex number in the complex plane.

$x = r \cdot cos(\theta)$

$y = r \cdot sin(\theta)$

$r = \sqrt{x^2 + y^2}$

$\theta = tan^{-1}(y/x)$ (Adjust quadrant!)

$r(cos \theta + i sin \theta)$

$r = \sqrt{a^2 + b^2}$

$\theta = tan^{-1}(b/a)$ (Adjust quadrant!)

$(r, \theta) = (-r, \theta + \pi)$

$(r, \theta) = (r, \theta + 2\pi k)$

$x = 2cos(7π/6) = -√3$, $y = 2sin(7π/6) = -1$