HYPERBOLIC FUNCTIONS
Synopsis :
1.
i) sinhx =
ex − e− x
2
ii) coshx =
ex + e− x
2
iii)tanhx =
iv) cothx =
v) sechx =
e x − e− x
ex + e− x
ex + e− x
ex − e− x
2
ex + e− x
vi) cosechx =
2
e − e− x
x
are called hyperbolic
functions.
Note : sin(ix) = isinhx, cos(ix) = coshx, tan(ix) = itanhx.
2.
cosh2x – sinh2x = 1
3.
1 – tanh2x = sech2x
4.
coth2x – 1 = cosech2x
5.
i) sinh (α + β) = sinh α cosh β + cosh α sinh β
ii) sinh (α – β) = sinh α cosh β – cosh α sinh β
iii)cosh (α + β) = cosh α cosh β + sinh α sinh β
iv) cosh (α – β) = cosh α cosh β – sinh α sinh β
6.
v) tanh (α + β) =
tanh α + tanh β
1 + tanh α tanh β
vi) tanh (α – β) =
tanh α − tanh β
1 − tanh α tanh β
i) sinh 2x = 2sinhx coshx
ii) cosh 2x = cosh2x + sinh2x
iii)cosh 2x = 2 cosh2 x – 1 or cosh2x =
iv) cosh 2x = 1 + sinh2x or 2sinh2x =
v) tanh 2x =
2 tanh x
1 + tanh2 x
1 + cosh 2x
2
cosh 2x − 1
2
Hyperbolic Functions
vi) sinh 3x = 3sinhx + 4sinh3x
vii)
cosh 3x = 4cosh3x – 3coshx
viii) tanh 3x =
7.
3 tanh x + tanh3 x
1 + 3 tanh2 x
Values of inverse hyperbolic functions as logarithms functions :
i) Sinh−1x = loge ( x + x 2 + 1)
ii) Cosh−1x = log( x + x 2 − 1), x ≥ 1
1+ x ⎞
⎟, x ∈ (-1, 1)
⎝ 1− x ⎠
iii) Tanh −1x = log⎛⎜
1
2
x + 1⎞
⎟, | x |> 1
⎝ x − 1⎠
iv) Coth−1x = log⎛⎜
1
2
⎛ 1± 1− x2
x
⎝
v) Sech−1x = log⎜⎜
⎞
⎟, 0 < x ≤ 1
⎟
⎠
⎛ 1+ 1+ x2
x
⎝
vi) Co sec h−1x = log⎜⎜
⎛ 1− 1+ x2
log⎜
⎜
x
⎝
⎞
⎟, x > 0 or
⎟
⎠
⎞
⎟, x < 0
⎟
⎠
2