7.3 Polar Coordinates

Polar coordinates give you a different way to locate points in a plane: instead of measuring horizontal and vertical distances (x, y), you measure how far a point is from the origin and what angle it makes with the positive x-axis. This system is far more natural for describing curves with circular or rotational structure, like spirals, roses, and cardioids.

Converting fluently between polar and rectangular coordinates is essential for this unit. Many curves that look nightmarish in rectangular form become elegant one-liners in polar form, and vice versa.

Polar Coordinate System

Plotting in polar coordinates

A point in polar coordinates is written as $(r, \theta)$, where:

• $r$ (the radial coordinate) measures the distance from the origin. While $r \geq 0$ in the simplest cases, negative $r$ values are allowed (more on that below).
• $\theta$ (the angular coordinate) measures the angle counterclockwise from the positive x-axis. It can be any real number.

The polar point $(r, \theta)$ corresponds to the rectangular point $(r\cos\theta,\, r\sin\theta)$.

A few things that trip people up:

• The origin is $(0, \theta)$ for any value of $\theta$. Since the distance is 0, the angle doesn't matter.
• Negative $r$ values flip the point to the opposite side. The point $(-r, \theta)$ is the same as $(r, \theta + \pi)$. You go $r$ units in the direction opposite to $\theta$.
• Non-unique representations. Unlike rectangular coordinates, every point has infinitely many polar representations. For example, $(3, \frac{\pi}{4})$ and $(3, \frac{\pi}{4} + 2\pi)$ are the same point, and so is $(-3, \frac{\pi}{4} + \pi)$.

Angles can be in degrees or radians. To convert:

• Degrees to radians: multiply by $\frac{\pi}{180}$
• Radians to degrees: multiply by $\frac{180}{\pi}$

Rectangular to polar transformations

Polar → Rectangular is straightforward:

1. $x = r\cos\theta$
2. $y = r\sin\theta$

Rectangular → Polar requires more care because of quadrant issues:

1. Find $r$: $r = \sqrt{x^2 + y^2}$
2. Find $\theta$: use $\theta = \tan^{-1}\!\left(\frac{y}{x}\right)$, then adjust for the correct quadrant.

The quadrant adjustments matter because $\tan^{-1}$ only returns values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$:

• If $x > 0$: $\theta = \tan^{-1}\!\left(\frac{y}{x}\right)$ (Quadrants I and IV are already covered)
• If $x < 0$ and $y \geq 0$: $\theta = \tan^{-1}\!\left(\frac{y}{x}\right) + \pi$ (Quadrant II)
• If $x < 0$ and $y < 0$: $\theta = \tan^{-1}\!\left(\frac{y}{x}\right) + \pi$ (Quadrant III)
• If $x = 0$ and $y > 0$: $\theta = \frac{\pi}{2}$
• If $x = 0$ and $y < 0$: $\theta = -\frac{\pi}{2}$

Note: for Quadrant III, adding $+\pi$ (rather than $-\pi$) gives you the standard angle in $(\pi, 2\pi)$ or equivalently you can use $-\pi$ to get the angle in $(-\pi, 0)$. Either convention works as long as the angle points to the correct quadrant.

Polar Curves and Equations

Graphing polar equations

To graph $r = f(\theta)$:

1. Choose $\theta$ values at regular intervals (increments of $\frac{\pi}{6}$ or $\frac{\pi}{4}$ work well).
2. Compute the corresponding $r$ for each $\theta$.
3. Plot each $(r, \theta)$ on a polar grid and connect the points smoothly.

Before plotting, check for symmetry (covered below) so you can reduce your work.

Common polar curve families:

• Cardioids: $r = a(1 \pm \cos\theta)$ or $r = a(1 \pm \sin\theta)$. Heart-shaped curves that pass through the origin. These are the special case of limaçons where $a = b$.
• Limaçons: $r = a \pm b\cos\theta$ or $r = a \pm b\sin\theta$. The shape depends on the ratio $\frac{a}{b}$:
  • $\frac{a}{b} < 1$: inner loop
  • $\frac{a}{b} = 1$: cardioid (no loop, cusp at origin)
  • $1 < \frac{a}{b} < 2$: dimpled, no loop
  • $\frac{a}{b} \geq 2$: convex (nearly oval)
• Rose curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$. If $n$ is odd, the curve has $n$ petals. If $n$ is even, it has $2n$ petals. Each petal has length $a$.
• Lemniscates: $r^2 = a^2\cos(2\theta)$ or $r^2 = a^2\sin(2\theta)$. Figure-eight shaped curves centered at the origin.
• Spirals: $r = a\theta$ (Archimedean spiral) or $r = ae^{b\theta}$ (logarithmic spiral). These wind outward from the origin.

Polar vs rectangular equations

Rectangular → Polar:

1. Replace $x$ with $r\cos\theta$ and $y$ with $r\sin\theta$.
2. Use $x^2 + y^2 = r^2$ when you see the sum of squares.
3. Simplify using trig identities.

Polar → Rectangular:

1. Multiply through by $r$ if needed so that terms like $r\cos\theta$ or $r\sin\theta$ appear (which you can replace with $x$ and $y$).
2. Replace $r^2$ with $x^2 + y^2$, $r\cos\theta$ with $x$, and $r\sin\theta$ with $y$.
3. Simplify.

This example shows why polar form is often cleaner: $r = 2\cos\theta$ is much simpler than $(x-1)^2 + y^2 = 1$ for graphing purposes.

Symmetry in polar curves

Symmetry tests let you reduce the amount of graphing work. There are three main tests:

• Symmetry about the polar axis (the line $\theta = 0$, i.e., the x-axis): Replace $\theta$ with $-\theta$. If the equation is unchanged, the curve is symmetric about the polar axis. Curves involving $\cos\theta$ often have this symmetry.
• Symmetry about the line $\theta = \frac{\pi}{2}$ (the y-axis): Replace $\theta$ with $\pi - \theta$. If the equation is unchanged, the curve is symmetric about this vertical line. Curves involving $\sin\theta$ often have this symmetry.
• Symmetry about the origin: Replace $r$ with $-r$ (or equivalently, replace $\theta$ with $\theta + \pi$). If the equation is unchanged, the curve has origin symmetry.

A caution: these are sufficient tests, not necessary ones. A curve can have a symmetry even if it fails the corresponding algebraic test, because of the non-uniqueness of polar representations. If a test passes, symmetry is guaranteed. If it fails, you can't be certain the symmetry is absent without further analysis.

Applications of Polar Coordinates

Connections to parametric equations and complex numbers

Polar coordinates connect naturally to parametric equations. Any polar curve $r = f(\theta)$ can be written parametrically as:

• $x(\theta) = f(\theta)\cos\theta$
• $y(\theta) = f(\theta)\sin\theta$

This is useful when you need to compute derivatives or arc lengths using parametric techniques from earlier in this unit.

Complex numbers also have a polar form. A complex number $z = a + bi$ can be written as $z = r(\cos\theta + i\sin\theta)$, where $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$. Euler's formula compresses this to $z = re^{i\theta}$, which makes multiplication and exponentiation of complex numbers much more intuitive: you multiply the moduli and add the angles.