Polar Coordinates and Polar Equations: Study Notes

10.1 Polar Coordinates and Polar Equations

Plotting Points Using Polar Coordinates

The polar coordinate system represents points in the plane using a distance from the origin (called the pole) and an angle from the positive x-axis (called the polar axis). A point is denoted as $(r, \theta)$, where $r$ is the directed distance and $\theta$ is the angle.

• If $r > 0$: The point lies on the terminal side of angle $\theta$ at a distance $r$ from the pole.

• If $r < 0$: The point lies on the ray opposite the terminal side of angle $\theta$ at a distance $|r|$ from the pole.

• If $r = 0$: The point is at the pole, regardless of $\theta$.

Polar grids are used for plotting polar coordinates, consisting of concentric circles and radial lines at standard angles.

Determining Different Representations of the Point $(r, \theta)$

Each point in polar coordinates can have multiple representations due to the periodic nature of angles and the sign of $r$.

• Same $r$, coterminal angles: $(r, \theta + 2k\pi)$ for any integer $k$ represents the same point.

• Opposite $r$, angle shifted by $\pi$: $(-r, \theta + (2k+1)\pi)$ for any integer $k$ also represents the same point.

• Point at the pole: $(0, \theta)$ for any $\theta$ is always at the origin.

Example: The point $(2, \frac{\pi}{3})$ can also be written as $(2, \frac{\pi}{3} + 2\pi)$ or $(-2, \frac{\pi}{3} + \pi)$.

Converting a Point from Polar Coordinates to Rectangular Coordinates

To convert from polar to rectangular coordinates, use the following relationships:

• $x = r \cos \theta$

• $y = r \sin \theta$

Example: For $(r, \theta) = (3, \frac{\pi}{4})$:

• $x = 3 \cos \frac{\pi}{4} = 3 \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}$

• $y = 3 \sin \frac{\pi}{4} = 3 \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}$

Converting a Point from Rectangular Coordinates to Polar Coordinates

Given a point $(x, y)$, the polar coordinates $(r, \theta)$ can be found as follows:

• $r = \sqrt{x^2 + y^2}$

• $\tan \theta = \frac{y}{x}$

Special cases for points lying along axes:

• $(a, 0)$: $r = a$, $\theta = 0$

• $(0, b)$: $r = b$, $\theta = \frac{\pi}{2}$

• $(-a, 0)$: $r = a$, $\theta = \pi$

• $(0, -b)$: $r = b$, $\theta = \frac{3\pi}{2}$

For points not on axes, determine the quadrant and adjust $\theta$ accordingly:

• Quadrant I: $\theta = \arctan\left(\frac{y}{x}\right)$

• Quadrant II: $\theta = \arctan\left(\frac{y}{x}\right) + \pi$

• Quadrant III: $\theta = \arctan\left(\frac{y}{x}\right) + \pi$

• Quadrant IV: $\theta = \arctan\left(\frac{y}{x}\right) + 2\pi$

Converting an Equation from Rectangular Form to Polar Form

A polar equation is an equation where the variables are $r$ and $\theta$. To convert from rectangular $(x, y)$ to polar $(r, \theta)$, use:

• $x = r \cos \theta$

• $y = r \sin \theta$

• $r^2 = x^2 + y^2$

Example: The circle $x^2 + y^2 = 9$ in polar form is $r^2 = 9$.

Converting an Equation from Polar Form to Rectangular Form

To convert from polar to rectangular form, use the same relationships:

• $x = r \cos \theta$

• $y = r \sin \theta$

• $r^2 = x^2 + y^2$

• $\tan \theta = \frac{y}{x}$

Example: The polar equation $r = 2 \cos \theta$ becomes $x^2 + y^2 = 2x$ in rectangular form.

Summary Table: Key Relationships Between Polar and Rectangular Coordinates

Rectangular	Polar
$x$	$r \cos \theta$
$y$	$r \sin \theta$
$r^2 = x^2 + y^2$	$r = \sqrt{x^2 + y^2}$
$\tan \theta = \frac{y}{x}$	$\theta = \arctan\left(\frac{y}{x}\right)$

Additional info: These notes cover the essential concepts for understanding and converting between polar and rectangular coordinates, as well as equations in both forms. Mastery of these conversions is foundational for further study in trigonometry, complex numbers, and analytic geometry.