# Hyperbolic functions

• ex = 1 + x/1! +x²/2! + x³/3! + … + xn / n! + … ∞
• e-x = 1 – x/1! +x²/2! – x³/3! + … + (-1)n xn / n! + … ∞
• sinhx = (ex – e-x) / 2 = x/1! +x²/3! + … + … ∞
• coshx = (ex + e-x) / 2 = 1 +x²/2! + … ∞
• tanhx = sinhx/ coshx = (ex – e-x)/ (ex + e-x)
• sechx = 1/ coshx = 2/ (ex + e-x)
• cosechx = 1/ sinhx = 2/ (ex + e-x)
• cothx = 1/ tanhx = (ex + e-x)/ (ex – 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]) = enx
• (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 = sinhxii) y = coshxiii) y = tanhxiv) y = cothxv) y = sechxvi) y = cosechx