8.3 Polar Coordinates

Polar Coordinate System

The polar coordinate system gives you a different way to locate points in a plane. Instead of moving left/right and up/down like rectangular coordinates $(x, y)$, you specify how far a point is from the origin ($r$) and what angle it makes with the positive x-axis ($\theta$). This system is especially useful for curves with circular or spiral shapes that would be messy to describe with $x$ and $y$.

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Characteristics of Polar Graphs

• Symmetry about the polar axis (the horizontal line through the pole): occurs when $f(\theta) = f(-\theta)$. Think of it like a mirror reflection across the horizontal axis.
• Symmetry about the line $\theta = \frac{\pi}{2}$: occurs when $f(\theta) = f(\pi - \theta)$. This is a reflection across the vertical axis.
• Symmetry about the pole (the origin): occurs when $f(\theta) = -f(\theta)$ or equivalently $f(\theta) = f(\theta + \pi)$. The graph looks the same if you rotate it 180°.
• Boundedness: A polar graph is bounded if $r$ has a finite maximum value. For example, $r = 3\cos(\theta)$ stays bounded since cosine never exceeds 1, so $r \leq 3$. Curves like spirals ($r = \theta$) are unbounded because $r$ grows without limit.
• Loops: Form when $r$ becomes negative for some values of $\theta$. A cardioid like $r = 1 + \cos(\theta)$ passes through the origin but doesn't loop, while $r = 1 + 2\cos(\theta)$ (a limaçon with an inner loop) does produce a loop because $r < 0$ for certain $\theta$ values.

Conversion of Coordinate Systems

Converting between polar and rectangular coordinates relies on basic right-triangle trig. The point $(r, \theta)$ sits at the end of a segment of length $r$ from the origin, at angle $\theta$ from the positive x-axis.

Polar to rectangular:

$x = r\cos(\theta)$

$y = r\sin(\theta)$

For example, the polar point $(4, \frac{\pi}{3})$ converts to $x = 4\cos(\frac{\pi}{3}) = 2$ and $y = 4\sin(\frac{\pi}{3}) = 2\sqrt{3}$, giving rectangular coordinates $(2, 2\sqrt{3})$.

Rectangular to polar:

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

$\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Watch the quadrant carefully. The arctangent function only returns values in $(-\frac{\pi}{2}, \frac{\pi}{2})$, so if the point is in Quadrant II or III, you need to add $\pi$ to get the correct angle. For example, the point $(-3, 3)$ has $r = 3\sqrt{2}$ and $\theta = \tan^{-1}(-1) + \pi = \frac{3\pi}{4}$ (not $-\frac{\pi}{4}$, which would land in Quadrant IV).

Angles in polar coordinates are typically expressed in radians.

Transformation of Polar Equations

Polar to rectangular: Replace polar expressions with their rectangular equivalents and simplify. A few substitutions come up constantly:

• $r\cos(\theta) = x$ and $r\sin(\theta) = y$
• $r^2 = x^2 + y^2$

For example, to convert $r = 4\cos(\theta)$: multiply both sides by $r$ to get $r^2 = 4r\cos(\theta)$, then substitute to get $x^2 + y^2 = 4x$. Completing the square gives $(x-2)^2 + y^2 = 4$, which is a circle centered at $(2, 0)$ with radius 2.

Rectangular to polar: Substitute $x = r\cos(\theta)$ and $y = r\sin(\theta)$ into the rectangular equation, then solve for $r$ as a function of $\theta$.

For example, $x^2 + y^2 = 9$ becomes $r^2 = 9$, so $r = 3$ (a circle of radius 3 centered at the origin).

Graphing Polar Equations

Graphing Techniques

Direct plotting (the go-to method):

1. Choose $\theta$ values, usually multiples of $\frac{\pi}{6}$ or $\frac{\pi}{4}$, from $0$ to $2\pi$.
2. Calculate the corresponding $r$ value for each $\theta$.
3. Plot each point $(r, \theta)$ on the polar plane. If $r$ is negative, plot the point in the opposite direction (add $\pi$ to the angle).
4. Connect the points with a smooth curve.

Rectangular conversion: Sometimes it's easier to convert to rectangular form first, recognize the shape (circle, line, etc.), and then sketch it. This works well for simpler equations like $r = 2\sec(\theta)$, which converts to $x = 2$ (a vertical line).

Symmetry and Periodicity

Testing for symmetry before graphing saves time because you may only need to plot half (or less) of the curve.

Symmetry tests:

1. Polar axis (horizontal): Replace $\theta$ with $-\theta$. If the equation is unchanged, the curve is symmetric about the polar axis.

2. The line $\theta = \frac{\pi}{2}$ (vertical): Replace $\theta$ with $\pi - \theta$. If the equation is unchanged, the curve is symmetric about the vertical line.

3. The pole (origin): Replace $r$ with $-r$. If the equation is unchanged, the curve is symmetric about the pole.

Periodicity: A polar curve repeats every $p$ radians if $f(\theta) = f(\theta + p)$. Common periods are $2\pi$ for basic trig functions, $\pi$ for functions involving $2\theta$, and $\frac{2\pi}{n}$ for functions involving $n\theta$. Rose curves like $r = \cos(3\theta)$ have period $\frac{2\pi}{3}$, so each petal repeats at that interval.

Effects of constants on polar graphs:

• Multiplying $f(\theta)$ by a constant $a$ scales the graph by $|a|$. If $a < 0$, the graph also reflects across the pole.
• Adding a constant $b$ to $\theta$ (writing $f(\theta + b)$) rotates the entire graph by angle $b$: counterclockwise when $b > 0$, clockwise when $b < 0$.

Common Polar Curves

Recognizing standard polar curve types helps you sketch graphs quickly:

• Circles: $r = a$, $r = a\cos(\theta)$, $r = a\sin(\theta)$
• Cardioids: $r = a \pm a\cos(\theta)$ or $r = a \pm a\sin(\theta)$ (heart-shaped, pass through the pole)
• Limaçons: $r = a \pm b\cos(\theta)$ where $a \neq b$ (with or without inner loop depending on whether $\frac{a}{b} < 1$)
• Rose curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$ ($n$ petals if $n$ is odd, $2n$ petals if $n$ is even)
• Spirals: $r = a\theta$ (Archimedean spiral, unbounded)

Polar graphs can reveal elegant patterns and shapes that rectangular coordinates would describe with much more complicated equations. Getting comfortable moving between both systems gives you flexibility in how you analyze and visualize curves.