hyperbolic functions

• e^x = 1 + x/1! +x²/2! + x³/3! + … + xⁿ / n! + … ∞
  
  
• e^-x = 1 - x/1! +x²/2! - x³/3! + … + (-1)ⁿ xⁿ / n! + … ∞
  
  
• sinhx = (e^x – e^-x) / 2 = x/1! +x²/3! + … + … ∞
  
  
• coshx = (e^x + e^-x) / 2 = 1 +x²/2! + … ∞
  
  
• tanhx = sinhx/ coshx = (e^x – e^-x)/ (e^x + e^-x)
  
  
• sechx = 1/ coshx = 2/ (e^x + e^-x)
  
  
• cosechx = 1/ sinhx = 2/ (e^x + e^-x)
  
  
• cothx = 1/ tanhx = (e^x + e^-x)/ (e^x – e^-x)
  
  
• sinh (-x) = - sinhx
  
  
• cosh (-x) = coshx
  
  
• tanh (-x) = - tanhx
  
  
• sechx (-x) = sechx
  
  
• cosech (-x) = - cosechx
  
  
• sinh (x ± y) = sinhx coshy ± coshx sinhy
  
  
• cosh (x ± y) = coshx coshy ± sinhx sinh
  
  
• tanh (x ± y) = (tanhx ± tanhy)/ (1 ± tanhx tanhy)
  
  
• sinh2x = 2 sinhx coshx = 2 tanhx/ (1 - tanh² x)
  
  
• cosh2x = cosh²x + sinh²x = (1+ tanh²x)/ (1 - tanh²x)
  
  
• tanh2x = 2tanhx/ (1 + tanh² x)
  
  
• sinh2x + cosh2x = (1 + tanhx) / (1 – tanhx)
  
  
• sinh3x = 3 sinhx + 4 sinh³x
  
  
• cosh 3x = 4 cosh³x – 3 coshx
  
  
• tanh 3x = (3 tanhx + tan³x)/ (1 + 3tanh²x)
  
  
• sinh (x + y) sinh(x - y) = sinh³x - sinh²y
  
  
• cosh (x + y) cosh (x - y) = cosh²x + sinh²y
  
  
• (coshx + sinhx)ⁿ = (cosh[nx] + sinh [nx]) = e^nx
  
  
• (coshx - sinhx)ⁿ = (cosh [nx] - sinh [nx]) = e^-nx
  
  
• cosh (2nx) + sinh (2nx) = [(1 + tanhx)/ (1 - tanhx)]ⁿ

Function	Domain	Range
sinhx	R	R
coshx	R	[1, ∞)
tanhx	R	(-1, 1)
cothx	R – {0}	R – [-1, 1]
cosechx	R – {0}	R – {0}
sechx	R	(0, 1]

 Graphs of Hyperbolic functions

i) y = sinhx

ii) y = coshx

iii) y = tanhx

iv) y = cothx

v) y = sechx

vi) y = cosechx