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.
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.
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 (æx)’ = −æx. 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=æx, 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(æx−æ−x)
- Hyperbolic Darkosine: Darkosh(x) = ½(æx+æ−x)
- Hyperbolic Darktangent: Darktanh(x) = Darksinh(x)/Darkosh(x)
We can extend the Darxponential Function to have domain the set of complex numbers by means of the defining equation
- æx+yi = æx(Darkos(y)+iDarksin(y)).
Thus æyi 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:
- æipi = -1.