are analogs of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle
just as the points (𝑐𝑜𝑠 𝑥, 𝑠𝑖𝑛 𝑥) form a circle with a unit radius, the points (𝑐𝑜𝑠ℎ 𝑥, 𝑠𝑖𝑛ℎ 𝑥) form the right half of the unit hyperbola

Hyperbolic Functions - Definitions in Exponentials

Function	Syntax	Definition	Complex Trigonometric Definitions	Derivative
Hyperbolic Sine	𝑠𝑖𝑛ℎ(𝑥)	$s inh (x) = \frac{e ^{x} - e ^{- x}}{2} = \frac{e ^{2 x} - 1}{2 e ^{x}} = \frac{1 - e ^{- 2 x}}{2 e ^{- x}}$	$s inh (x) = - i s in (i x)$	$s in h^{'} (x) = cos h (x)$
Hyperbolic Cosine	𝑐𝑜𝑠ℎ(𝑥)	$cos h (x) = \frac{e ^{x} + e ^{- x}}{2} = \frac{e ^{2 x} + 1}{2 e ^{x}} = \frac{1 + e ^{- 2 x}}{2 e ^{- x}}$	$cos h (x) = cos (i x)$	$cos h^{'} (x) = s inh (x)$
Hyperbolic Tangent	𝑡𝑎𝑛ℎ(𝑥)	$t anh (x) = \frac{s inh ( x )}{cos h ( x )} = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}} = \frac{e ^{2 x} - 1}{e ^{2 x} + 1}$	$t anh (x) = - i t an (i x)$	$t an h^{'} (x) = 1 - t an h^{2} (x) = sec h^{2} (x)$
Hyperbolic Cotangent	𝑐𝑜𝑡ℎ(𝑥)	$coth(x) = \frac{cosh(x)}{sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}} = \frac{e^{2x} + 1}{e^{2x} - 1} \quad \text{for x ≠ 0}$	$co t h (x) = i co t (i x)$	$co t h^{'} (x) = - csc h^{2} (x)$
Hyperbolic Secant	𝑠𝑒𝑐ℎ(𝑥)	$sec h (x) = \frac{1}{cos h ( x )} = \frac{2}{e ^{x} + e ^{- x}} = \frac{2 e ^{x}}{e ^{2 x} + 1}$	$sec h (x) = sec (i x)$	$sec h^{'} (x) = - sec h (x) t anh (x)$
Hyperbolic Cosecant	𝑐𝑠𝑐ℎ(𝑥)	$csch(x) = \frac{1}{sinh(x)} = \frac{2}{e^x - e^{-x}} = \frac{2e^x}{e^{2x} - 1} \quad \text{for x ≠ 0}$	$csc h (x) = i csc (i x)$	$csc h^{'} (x) = - csc h (x) co t h (x)$

／var／log marcus chiu