Polar Coordinates & Hyperbolic Functions

We can easily graph any polar formula $r=f(\theta)$ following the following steps:

1. Plot $r\ -\ \theta$ graph. If the function involves trigo functions, we can simply translate its graph. The height at a certain point is the distance from the origin.
2. Then we can simply trace the graph from the left, and plot the points according to the distance from the origin at each angle. Using the eqations above, we can also find the Cartesian coordinates.
3. If the graph is symmetric, we can simply copy the other half.

The following identities hold for hyperbolic functions:

1. $\cosh^2x-\sinh^2x=1$
2. $\cosh 2x=\cosh^2x+\sinh^2x$
3. $\sinh 2x=2\sinh x\cosh x$

The derivatives of hyperbolic functions are:

1. $\frac{d}{dx}\sinh x=\cosh x$
2. $\frac{d}{dx}\cosh x=\sinh x$
3. $\frac{d}{dx}\sinh^{-1}x=\frac{1}{\sqrt{x^2+1}}$
4. $\frac{d}{dx}\cosh^{-1}x=\frac{1}{\sqrt{x^2-1}}$