Hyperbolic Functions

Cuthbert Nyack

sinh x = (e^x - e^-x)/2
cosh x = (e^x + e^-x)/2
tanh x = sinh x/cosh x

e^x = cosh x + sinh x
e^-x = cosh x - sinh x

cosh² x - sinh² x = 1
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
sinh(x ± y) = sinh x cosh y ± cosh x sinh y

d/dx sinh ax = a cosh ax
d/dx cosh ax = a sinh ax
d/dx tanh ax = a sech² ax

d/dx asinh x/a = (x² + a²)^-½
d/dx acosh x/a = (x² - a²)^-½
d/dx atanh x/a = a(a² - x²)^-1

ò (sinh ax) dx = (1/a) cosh ax
ò (cosh ax) dx = (1/a) sinh ax
ò (tanh ax) dx = (1/a) log_e cosh ax

for |x| < 1
sinh x = x + x³/3! + x⁵/5! + ...
cosh x = 1 + x²/2! + x⁴/4! + ...

Return to main page
Return to page index