In calculus, we are taught that radians are **the** best unit of measurement for angles. But technically, it is more accurate to say that radians are **a** best unit of measurement for angles. There is one other unit which is exactly as good: *The Dark Radian*.

Whereas a radian is defined so that one revolution equals 2pi radians, a Dark Radian is defined so that one revolution equals −2pi radians. Thus, 1 Dark Radian is −1 radians, and 1 radian is −1 Dark Radian.

## Dark Trigonometry

We define the *dark trigonometric functions* as follows, where x is a real number:

- Darksine. We define Darksin(x) = sin(−x), where sin is the usual sine function. Thus, Darksin(x) is the “opposite/hypotenuse” corresponding to an angle of x Dark Radians in a right triangle.
- Equivalently, Darksin(x) is “opposite/hypotenuse” corresponding to an angle of x radians in a wrong triangle. A wrong triangle is, of course, a triangle one of whose angles is pi/2 Dark Radians.

- Darkosine. We define Darkos(x) = cos(−x), similar to the above.
- Darktangent. We define Darktan(x) = tan(−x), similar to the above.

The dark trigonometric functions differentiate as follows: (is *differentiate* an ergative verb?)

- (Darksin(x))’ = −Darkcos(x).
- (Darkcos(x))’ = Darksin(x).
- (Darktan(x))’ = −1/Darkcos²(x) (or −Darksec²(x) in some dialects).

The Dark trig functions have inverse functions, when suitably restricted. These we call Darkarcsine, Darkarccosine, and Darkarctangent. Their derivatives are left as an exercise to the reader.

## Dark Exponentials

It is well-known that there is one base of exponents preferred above all others. I am speaking, of course, of the Dark Exponential Base, æ, also known as “Dark e”. It has a numerical value of approximately 0.36787. It is related to its holier sister, e, by the formula æ=1/e.

The reason æ is preferred is because it is the unique number such that (æ^{x})’ = −æ^{x}. In other words, it is the base of an exponential function which is its own negative derivative. Thus, the unique (up to linear combinations) solutions to the differential equation y””=y are y=Darksin(x), y=Darkos(x), y=æ^{x}, and y=æ^{−x}, explaining why these four Dark Functions play such a key role in the deep laws and corruptions of our Dark universe.

The Darksponential can be used to define the Dark-Hyperbolic Trigonometric functions:

- Hyperbolic Darksine: Darksinh(x) = ½(æ
^{x}−æ^{−x}) - Hyperbolic Darkosine: Darkosh(x) = ½(æ
^{x}+æ^{−x}) - Hyperbolic Darktangent: Darktanh(x) = Darksinh(x)/Darkosh(x)

## Darkomplex Analysis

We can extend the Darxponential Function to have domain the set of complex numbers by means of the defining equation

- æ
^{x+yi}= æ^{x}(Darkos(y)+**i**Darksin(y)).

Thus æ^{yi} is the point on the Darkomplex unit circle with an angle of y Dark Radians from the positive x-axis.

Plugging in x=0, y=pi yields the infamous Dark-Euler’s Formula:

- æ
^{ipi}= -1.