Formulae:
- sinh-1 x = loge (x + √(x² + 1))
- cosh-1 x = loge (x + √(x² – 1)) for x ≥ 1
- tanh-1 x = ½ log [(1 +x)/ (1 – x)] for x ϵ (-1, 1)
- coth-1 x = ½ log [(1 +x)/ (1 – x)] for |x| > 1
- sech-1 x = loge (1 + √ (1 – x²)/ x) for x ϵ (0, 1]
- cosech-1 x = loge (1 + √ (1 + x²)/ x) for x > 0
= loge (1 – √ (1 + x²)/ x) for x < 0 - sinh-1 x = cosh-1 √(x² + 1) = cosech-1 (1/ x) = tanh-1 (x / √ (1 + x²))
- cosh-1 x = sinh-1 √(x² – 1) = sech-1 (1/ x) = tanh-1 (√ (1 + x²) / x)
Function |
domain |
Range |
sinh-1 x |
R | R |
cosh-1 x | [1, ∞) |
[0, ∞) |
tanh-1 |
(-1, 1) | R |
coth-1 x | R – [-1, 1] |
R – {0} |
sech-1 x |
(0,1] | [0, ∞) |
cosech-1 x | R – {0} |
R – {0} |
Graphs of inverse hyperbolic functions:
i) y = sinh-1 xii) y = cosh-1 x
iii) y = tanh-1 x
iv) y = sech-1 x
v) y = cosech-1 xvi) y = coth-1 x