Polar coordinates offer a unique way to represent points and curves in a two-dimensional plane. Instead of using x and y, we use distance from the origin (r) and angle from the x-axis (θ). This system is particularly useful for describing circular and spiral shapes.

Converting between polar and rectangular coordinates is a key skill. It allows us to choose the most convenient system for a given problem. Polar coordinates shine when dealing with circular motion, periodic functions, and certain types of symmetry.

Polar Coordinate System

Plotting in polar coordinates

• Polar coordinates represent a point's position using distance from the origin (r) and angle from the positive x-axis (θ)
  • r is the radial coordinate measures distance from the origin ($r \geq 0$)
  • θ is the angular coordinate measures angle counterclockwise from the positive x-axis (θ can be any real number)
• The polar point (r, θ) corresponds to the rectangular point $(r \cos \theta, r \sin \theta)$
• The origin is represented by (0, θ) for any value of θ since the distance from the origin is 0
• Negative r values represent points located in the opposite direction from θ (r units from the origin at an angle of θ + π)
• Angles can be expressed in degrees or radians
  • To convert from degrees to radians, multiply by $\frac{\pi}{180}$ (1 degree = $\frac{\pi}{180}$ radians)
  • To convert from radians to degrees, multiply by $\frac{180}{\pi}$ (1 radian = $\frac{180}{\pi}$ degrees)
• A polar grid is used to visualize and plot points in the polar coordinate system

Rectangular to polar transformations

• To convert from polar coordinates (r, θ) to rectangular coordinates (x, y):
  1. $x = r \cos \theta$ projects the polar point onto the x-axis
  2. $y = r \sin \theta$ projects the polar point onto the y-axis
• To convert from rectangular coordinates (x, y) to polar coordinates (r, θ):
  1. $r = \sqrt{x^2 + y^2}$ calculates the distance from the origin using the Pythagorean theorem
  2. $\theta = \tan^{-1}(\frac{y}{x})$ finds the angle using the arctangent function, with quadrant adjustments based on the signs of x and y
  • If $x > 0$, then $\theta = \tan^{-1}(\frac{y}{x})$ (point is in quadrant I)
  • If $x < 0$ and $y \geq 0$, then $\theta = \tan^{-1}(\frac{y}{x}) + \pi$ (point is in quadrant II)
  • If $x < 0$ and $y < 0$, then $\theta = \tan^{-1}(\frac{y}{x}) - \pi$ (point is in quadrant III)
  • If $x = 0$ and $y > 0$, then $\theta = \frac{\pi}{2}$ (point is on the positive y-axis)
  • If $x = 0$ and $y < 0$, then $\theta = -\frac{\pi}{2}$ (point is on the negative y-axis)

Polar Curves and Equations

Graphing polar equations

• To graph a polar equation $r = f(\theta)$, create a table of values for θ and calculate the corresponding r values
  1. Choose a set of θ values (usually in increments of $\frac{\pi}{6}$ or $\frac{\pi}{4}$)
  2. Calculate the corresponding r values using the polar equation
  3. Plot the points (r, θ) on the polar coordinate system and connect them smoothly
• Common polar curve shapes include:
  • Cardioids: $r = a \pm b \cos \theta$ or $r = a \pm b \sin \theta$ (heart-shaped curves)
  • Limaçons: $r = a \pm b \cos \theta$ or $r = a \pm b \sin \theta$, where $a > b$ for inner loops (snail-shaped curves)
  • Roses: $r = a \cos (n\theta)$ or $r = a \sin (n\theta)$, where n is an integer (flower-shaped curves with n petals)
  • Lemniscates: $r^2 = a^2 \cos (2\theta)$ or $r^2 = a^2 \sin (2\theta)$ (figure-eight shaped curves)
  • Spirals: $r = a\theta^{\frac{1}{n}}$, where n is an integer (curves that wind around the origin)
• The polar area formula can be used to calculate the area enclosed by a polar curve

Polar vs rectangular equations

• To convert a rectangular equation to polar form:
  1. Substitute $x = r \cos \theta$ and $y = r \sin \theta$ into the rectangular equation
  2. Simplify the equation using trigonometric identities to express it in terms of r and θ
• To convert a polar equation to rectangular form:
  1. Substitute $r \cos \theta$ for x and $r \sin \theta$ for y in the polar equation
  2. Simplify the equation using trigonometric identities to express it in terms of x and y
• Some equations are easier to express and graph in polar form than in rectangular form ($r = 1 + \cos \theta$ cardioid vs $(x^2 + y^2 - 2x)^2 = 4(x^2 + y^2)$)

Symmetry in polar curves

• Symmetry about the polar axis (θ = 0):
  • If $r = f(\theta)$ is symmetric about the polar axis, then $f(\theta) = f(-\theta)$ (curve is unchanged when reflected across the polar axis)
• Symmetry about the vertical line (θ = π/2):
  • If $r = f(\theta)$ is symmetric about the vertical line, then $f(\theta) = f(\pi - \theta)$ (curve is unchanged when reflected across the vertical line)
• Rotational symmetry:
  • If $r = f(\theta)$ has rotational symmetry of order n, then $f(\theta) = f(\theta + \frac{2\pi}{n})$ (curve repeats itself every $\frac{2\pi}{n}$ radians)
• Symmetry tests for polar equations:
  • $r(\theta) = r(-\theta)$ implies symmetry about the polar axis (reflection across θ = 0)
  • $r(\theta) = -r(\theta + \pi)$ implies symmetry about the origin (half-turn rotation)
  • $r(\theta) = r(\theta + \pi)$ implies symmetry about the vertical line (reflection across θ = π/2)

Applications of Polar Coordinates

Parametric equations and complex numbers

• Polar coordinates can be used to represent parametric equations, where both x and y are expressed in terms of a parameter t
• Complex numbers can be represented in polar form, connecting algebra and geometry
• Euler's formula relates complex exponentials to trigonometric functions, bridging polar and rectangular representations