Precalculus

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Rules of Thumb

Polar coordinates are of the form $(r,\theta)$ where $r$ is distance and is direction in degrees/radians
Although they exist in separate planes, polar coordinates and cartesian (including imaginary) coordinates are convertible between each other using trigonometry
“The only thing you need in order to completely evaluate a right triangle is an angle and a side” - Mr. Noble


What to Review

Trigonometry
Solving right triangles


Converting Between Cartesian and Polar Coordinates

If you have an ordered pair in the Cartesian plane, like $(3,4)$, then the $x$ value is the length of the horizontal leg of a right triangle and the $y$ value is the length of the vertical length of the same triangle. The hypotenuse, or in this case $r$, can then be calculated using the pythagorean theorem.
To calculate $\theta$, use appropriate trigonometry.


Graphs

Sometimes, the $\theta$ in the polar coordinate can be replaced by an operator such as $\cos\theta$. This results in lit graphs such as:
Limaçon

Limacon

Lemniscate

Lemniscate

Rose

Rose
Rose With 3 Petals

Note: a rose has the equation $r=\cos(k\theta)$; if $k$ is odd, then the rose will have $k$ petals; if $k$ is even, then the rose will have $2k$ petals.
Spiral

Spiral

De Moivre’s Theorem

A theorem useful for finding powers of complex numbers
Coordinates must be in polar form
$r(\cos\theta+i\cdot\sin\theta)=rcis\theta$ (archaic)
$r^n[\cos(n\theta)+i\cdot\sin(n\theta)]=z^n=r(\cos+i\cdot\sin)^n$
$z^n$ is the polar form of a complex number