How to Calculate the Hyperbolic Tangent (tanh)

This page works out the hyperbolic tangent of a number, and the hyperbolic sine and cosine it is built from. Enter x (a plain number, not an angle in degrees); you get tanh x, sinh x and cosh x to six decimal places.

Formula

sinh x = (eˣ − e⁻ˣ) ÷ 2, cosh x = (eˣ + e⁻ˣ) ÷ 2 and tanh x = sinh x ÷ cosh x = (eˣ − e⁻ˣ) ÷ (eˣ + e⁻ˣ).

• x: any real number (the calculator accepts −300 to 300)
• e ≈ 2.718282, the base of natural logarithms

Worked Example

Take x = 0.5.

1. e^0.5 = 1.648721 and e^−0.5 = 0.606531
2. sinh 0.5 = (1.648721 − 0.606531) ÷ 2 = 0.521095
3. cosh 0.5 = (1.648721 + 0.606531) ÷ 2 = 1.127626
4. tanh 0.5 = 0.521095 ÷ 1.127626 = 0.462117

How to read the result

tanh x always lies between −1 and 1. It is close to x for small x, and it flattens towards 1 (or −1) as x grows, which is why it is used as an activation function in neural networks and in the velocity of a falling body with air drag. cosh x is never below 1, and cosh² x − sinh² x = 1 for every x.

Limits and common mistakes

• x is not an angle: there is no degrees mode for hyperbolic functions.
• For x above about 20, tanh x rounds to 1.000000 at six decimals.
• Do not confuse tanh with tan; tan uses an angle and repeats every π.

Frequently Asked Questions

What is tanh 0?

0, because sinh 0 = 0 and cosh 0 = 1.

What is tanh 1?

About 0.761594.

Is tanh an odd function?

Yes: tanh(−x) = −tanh x. sinh is odd too, while cosh is even.

How do I find the inverse, artanh?

artanh y = ½ ln((1 + y) ÷ (1 − y)) for −1 < y < 1.

in x | Value x | - | 0.5 | -300 | 300
out th | tanh x | - | (exp(x) - exp(-x)) / (exp(x) + exp(-x)) | 6
out sh | sinh x | - | (exp(x) - exp(-x)) / 2 | 6
out ch | cosh x | - | (exp(x) + exp(-x)) / 2 | 6