Generalized Trigonometric Functions
This page documents how trigonometric functions generalize to elliptical trigonometry and hyperbolic trigonometry.
Preliminaries
Recall the power series expansions of the conventional trigonometric functions and their inverses.
Trigonometric Functions
For the conventional trigonometric functions, we have the following series:
In particular, note how the series for
For example, it follows directly that:
And also:
Inverse Trigonometric Functions
Furthermore, we have the following series, which converge for for
For example, it follows directly that:
Generalization
Definitions
The power series above suggest a possible generalization:
Note that from these power series, it follows immediately:
As in conventional trigonometry, we can define the tangent function as:
The power series are sufficient to define
The inverses of
(Note: I first encountered these formulations in McRae.)
Identities
The generalized functions obey the following generalized "Pythagorean" identity:
The following angle addition/subtraction identities hold:
The following angle multiplication/division identities hold:
As special cases of the above, we have:
Derivatives
The derivatives of
Plots
By plotting

By plotting

By plotting

Polar Coordinates
Let us consider the following parametric equation:
With the aid of the generalized "Pythagorean" identity we can derive the implicit form by eliminating
By setting

As can be seen in the plot, resulting shapes are:
- Ellipses for
- Paralell Line Pair for
- Hyperbolas for
Demos
Angles
The demo below illustrates the generalized concept of angles for varying values of
Polar Coordinates
The demo below shows the real (solid) and imaginary (dashed) unit circles for varying values of