10.3 Polar Coordinates

The polar coordinate system offers a different way to represent points in two dimensions. Instead of measuring horizontal and vertical distances (like rectangular coordinates), you describe a point by its distance from the origin and the angle it makes with the positive x-axis. This system is especially handy for curves that involve rotation or radial symmetry, like circles, spirals, and flower-shaped graphs.

Converting between polar and rectangular coordinates is a core skill in this section. You'll also need to transform equations between the two systems and recognize the shapes that common polar equations produce.

Polar Coordinate System

Plotting in polar coordinates

A polar coordinate is written as $(r, \theta)$, where $r$ is the distance from the origin (called the pole) and $\theta$ is the angle formed with the positive x-axis (called the polar axis).

Angles can be measured in degrees or radians:

• Positive angles are measured counterclockwise from the positive x-axis (e.g., 45°, 90°)
• Negative angles are measured clockwise (e.g., -30°, -120°)

To plot a polar point like $(2, 60°)$:

1. Start at the origin and rotate 60° counterclockwise from the positive x-axis.
2. Move outward 2 units along that direction.
3. Mark the point.

A polar grid makes this easier. It consists of concentric circles (showing distance from the origin) and radial lines (showing common angles), so you can quickly locate any $(r, \theta)$ pair.

One thing that can be confusing: unlike rectangular coordinates, polar representations aren't unique. The point $(2, 60°)$ is the same as $(2, 420°)$ or $(-2, 240°)$. A negative $r$ value means you go in the opposite direction of the angle.

Conversion of coordinate systems

Polar to rectangular: Given $(r, \theta)$, find $(x, y)$ using:

$x = r\cos(\theta)$ $y = r\sin(\theta)$

Example: Convert $(3, \frac{\pi}{4})$ to rectangular.

• $x = 3\cos(\frac{\pi}{4}) = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}$
• $y = 3\sin(\frac{\pi}{4}) = 3 \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}$
• Rectangular form: $(\frac{3\sqrt{2}}{2},\ \frac{3\sqrt{2}}{2})$

Rectangular to polar: Given $(x, y)$, find $(r, \theta)$ using:

$r = \sqrt{x^2 + y^2}$ $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Watch out: the $\tan^{-1}$ formula only gives you an angle in Quadrants I or IV. If the point is in Quadrant II or III (meaning $x < 0$), you need to add $\pi$ to get the correct angle.

Example: Convert $(2, 2)$ to polar.

• $r = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$
• $\theta = \tan^{-1}(\frac{2}{2}) = \tan^{-1}(1) = \frac{\pi}{4}$
• Both coordinates are positive (Quadrant I), so no adjustment needed.
• Polar form: $(2\sqrt{2},\ \frac{\pi}{4})$

Polar Equations and Graphs

Transformation of polar equations

You can convert equations between polar and rectangular form using the same relationships: $x = r\cos(\theta)$, $y = r\sin(\theta)$, and the identity $x^2 + y^2 = r^2$.

Polar to rectangular example: Convert $r = 2\cos(\theta)$.

1. Multiply both sides by $r$: $r^2 = 2r\cos(\theta)$
2. Substitute $r^2 = x^2 + y^2$ and $r\cos(\theta) = x$: $x^2 + y^2 = 2x$
3. This is a circle in rectangular form (you can complete the square to get $(x-1)^2 + y^2 = 1$, a circle centered at $(1, 0)$ with radius 1).

Rectangular to polar example: Convert $x^2 + y^2 = 4$.

1. Substitute $x^2 + y^2 = r^2$: $r^2 = 4$
2. Take the positive root: $r = 2$
3. This is simply a circle of radius 2 centered at the origin. Much cleaner in polar form.

Graphing polar equations

To graph a polar equation by hand:

1. Choose $\theta$ values across the interval $[0, 2\pi]$ (or $[0, \pi]$ if symmetry allows).
2. Calculate the corresponding $r$ for each $\theta$.
3. Plot each $(r, \theta)$ point on a polar grid.
4. Connect the points smoothly.

Here are the common polar curve types you should recognize:

Polar vs rectangular representations

Some curves are far simpler in polar form. A cardioid like $r = a(1 + \cos\theta)$ would be a mess to write in rectangular coordinates, but in polar form it's a single clean equation.

Symmetry tests help you graph more efficiently:

• Symmetric about the polar axis (x-axis): Replace $\theta$ with $-\theta$. If the equation is unchanged, the graph is symmetric about the horizontal axis.
• Symmetric about the line $\theta = \frac{\pi}{2}$ (y-axis): Replace $\theta$ with $\pi - \theta$. If unchanged, the graph is symmetric about the vertical axis.
• Symmetric about the pole (origin): Replace $r$ with $-r$ (or equivalently, replace $\theta$ with $\theta + \pi$). If unchanged, the graph is symmetric about the origin.

Petal count for rose curves is a detail worth memorizing:

• If $n$ is odd, $r = a\cos(n\theta)$ has exactly $n$ petals. For example, $r = \cos(3\theta)$ has 3 petals.
• If $n$ is even, the curve has $2n$ petals. So $r = \cos(2\theta)$ has 4 petals, not 2.