How to use it
- Choose sinh, cosh, tanh… for the direct functions or Inverse for arsinh, arcosh, artanh and arcoth.
- Enter x. Negative numbers and decimals are fine.
- Read the results. A function that is not defined at x, for example coth 0 or arcosh 0.5, is left out of the list.
Formulas
- sinh x = (eˣ − e⁻ˣ) / 2, cosh x = (eˣ + e⁻ˣ) / 2
- tanh x = sinh x / cosh x, coth x = cosh x / sinh x (x ≠ 0)
- sech x = 1 / cosh x, csch x = 1 / sinh x (x ≠ 0)
- arsinh x = ln(x + √(x² + 1))
- arcosh x = ln(x + √(x² − 1)), x ≥ 1
- artanh x = ½ ln((1 + x) / (1 − x)), |x| < 1
- arcoth x = ½ ln((x + 1) / (x − 1)), |x| > 1
Worked example
Take x = ln 2 ≈ 0.693147. Then eˣ = 2 and e⁻ˣ = 0.5, so sinh x = (2 − 0.5)/2 = 0.75, cosh x = (2 + 0.5)/2 = 1.25 and tanh x = 0.75/1.25 = 0.6. The reciprocals are coth 1.66666667, sech 0.8 and csch 1.33333333. Check: 1.25² − 0.75² = 1.5625 − 0.5625 = 1.
In inverse mode, x = 0.6 gives artanh 0.6 = ½ ln(1.6/0.4) = ½ ln 4 = 0.69314718, which is ln 2 again.
Tips
- Identities to check results: cosh² − sinh² = 1, 1 − tanh² = sech², and sinh(2x) = 2 sinh x cosh x.
- cosh is even, sinh and tanh are odd: cosh(−x) = cosh x, sinh(−x) = −sinh x.
- A hanging cable follows y = a cosh(x/a), the catenary, which is where cosh shows up in engineering.