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 = rcos(θ), y = rsin(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)

Cartesian: (x, y), uses perpendicular axes | Polar: (r, θ), uses distance and angle from origin

Positive r: Point is in the direction of θ | Negative r: Point is in the opposite direction of θ

Real axis: Represents the real part of a complex number | Imaginary axis: Represents the imaginary part of a complex number

Rectangular: a + bi | Polar: r(cos θ + i sin θ)

Magnitude: Distance from origin | Argument: Angle from positive real axis

Degree: Unit of angle, 360° in a circle | Radian: Unit of angle, 2π radians in a circle

x: Cartesian coordinate | r*cos(θ): Equivalent in polar coordinates

y: Cartesian coordinate | r*sin(θ): Equivalent in polar coordinates

tan⁻¹(y/x): Gives angle in quadrants I or IV | θ: Requires quadrant adjustment for correct angle

Complex number: Has real and imaginary parts (a + bi) | Real number: Only has a real part

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

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

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

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

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

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

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

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

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

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