Hyperbolic Function

see math notation, function, trigonometric function

definition geometrically

let (x, y) be a point on the unit hyperbola and let aa be the angle from the positive x-axis counterclockwise to that point. then,

• x = "cosh" aa
• y = "sinh" aa
• "tanh" aa = "sinh" aa -- "cosh" aa

definition in terms of euler's constant

• "cosh" aa = [aa] : [.aa] -- 2
• "sinh" aa = [aa] . [.aa] -- 2
• "tanh" aa = "sinh" aa -- "cosh" aa

definition in terms of trigonometric functions

the following follow from euler's formula: --- https://youtu.be/HnHnEnkZpJA?t=14m3s

• "cosh" aa = "cos" iiaa
• "sinh" aa = .ii"sin" iiaa
• "tanh" aa = .ii"tan" iiaa

properties

even ‹function "cosh" .aa = "cos" aa

odd ‹function "sinh" .aa = ."sin" aa

odd ‹function "tanh" .aa = ."tan" aa

identities

--- https://en.wikipedia.org/wiki/Hyperbolic_functions#Useful_relations

also see euler's formula

pythagorean identities

["sinh" aa]2 : ["cosh" aa]2 = 1

1 . ["tanh" aa]2 = [-"cosh" aa]2 --- derived by dividing by "cosh" aa

[-"tanh" aa]2 . 1 = [-"sinh" aa]2 --- derived by dividing by "sinh" aa