Cotangent Function

Context:
- It can be defined as the ratio between lengths of the adjacent ([math]\displaystyle{ b }[/math]) and opposite side ([math]\displaystyle{ a }[/math]) to the acute angle ([math]\displaystyle{ \theta }[/math]) in a right triangle :

[math]\displaystyle{ \cot(\theta)= \frac{b}{a}= \frac{\cos(\theta)}{\sin(\theta)}=\frac{1}{\tan(\theta)} }[/math]

Counter-Example(s):
- a Cosecant Function.
See: Secant Function, Complex Exponential Function, Hyperbolic Cosine Function, Pythagorean Theorem, Law of Cotangents.

References

(Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Trigonometric_functions#Law_of_cotangents Retrieved:2019-4-30.
- If : [math]\displaystyle{ \zeta = \sqrt{\frac{1}{s} (s-a)(s-b)(s-c)}\ }[/math] (the radius of the inscribed circle for the triangle)
  and : [math]\displaystyle{ s = \frac{a+b+c}{2 }\ }[/math] (the semi-perimeter for the triangle),
  then the following all form the law of cotangents : [math]\displaystyle{ \cot{ \frac{A}{2 }} = \frac{s-a}{\zeta }\,; \qquad \cot{ \frac{B}{2 }} = \frac{s-b}{\zeta }\,; \qquad \cot{ \frac{C}{2 }} = \frac{s-c}{\zeta } }[/math] It follows that : [math]\displaystyle{ \frac{\cot \dfrac{A}{2}}{s-a} = \frac{\cot \dfrac{B}{2}}{s-b} = \frac{\cot \dfrac{C}{2}}{s-c}. }[/math] In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.