All my math notes, now in Markdown.

You should be able to

- Plot polar points
- Find equivalent polar points to a given point
- Convert from polar to rectangular and rectangular to polar

A polar coordinate system is when points can be graphed from a directed distance and angle. The distance from the pole is called the radical coordinate, or **radius**, and the angle is called the angular coordinate, or **azimuth / polar angle**.

The **Pole**: Point $O$

The **Polar Axis**: ray from point $O$ along positive x-axis

The **Polar Coordinates**: $(r, \theta)$

$r$ is directed distance from $O$

$\theta$ is directed angle from polar axis

**Polar graphs** are circular. The point $(4, \frac{4 \pi}{3})$ is plotted below. (could also be labeled as $(-4, \frac{\pi}{3})$)

If the point $P$ has polar coordinates $(r, \theta)$, **all other polar coordinates** of $P$ must have the form $(r, \theta + 2 \pi n)$ or $(-r, \theta + \pi + 2 \pi n)$ where $n$ is any integer.

If the point $P$ has polar coordinates $(r, \theta)$ and rectangular coordinates $(x, y)$, then

$x = r\cos{\theta}, \qquad y = r \sin{\theta}$

$\tan{\theta} = \frac{y}{x}, \qquad x^2 + y^2 = r^2$

Graph $r=4\cos\theta$, then write the polar equation in rectangular form.

- Find two polar coordinate pairs for the rectangular coordinate point $(-1,1)$
- Write $r=4\sec\theta$ in rectangular form

- For $P(3,\frac{2\pi}{3})$
- Plot the point
- Convert to exact rectangular coordinates

- For $Q(-2,\frac \pi 4)$
- Plot the point
- Convert to exact rectangular coordinates

- For $R(2, \frac \pi 6$
- Find another polar coordinate for the point $R$ on the interval $[0,2\pi)$
- Find
*all*other polar points.

- Convert $(2\sqrt 3,-2)$ to polar form.
- Convert $r=-3\sin\theta$ to rectangular form.
- Convert $y=5$ to polar form.
- Convert $r=2\sin\theta-4\cos\theta$ to rectangular form.
- Convert $(x-1)^2+(y+4)^2 = 17$ to polar form.
- The location, given in polar coordinates, of two planes approaching an airport are $(4\ mi,12°)$ and $(2\ mi,72°)$. Find the distance between them.