1. What is Hyperbolic Tangent?

The hyperbolic tangent (tanh) is a hyperbolic function that is the ratio of the hyperbolic sine (sinh) to the hyperbolic cosine (cosh). It is analogous to the ordinary trigonometric tangent function but for a hyperbola rather than a circle.

2. How Does the Calculator Work?

The calculator uses the hyperbolic tangent formula:

Where:

• \( \sinh(x) \) — Hyperbolic sine of x
• \( \cosh(x) \) — Hyperbolic cosine of x

Explanation: The function takes any real number x and returns a value between -1 and 1. It's an odd function, meaning tanh(-x) = -tanh(x).

3. Applications of Tanh Function

Details: The tanh function is widely used in physics, engineering, and machine learning (as an activation function in neural networks). It describes the shape of a hanging cable (catenary) and appears in solutions to certain differential equations.

4. Using the Calculator

Tips: Enter the sinh(x) and cosh(x) values. The cosh(x) value cannot be zero as this would make tanh(x) undefined. The result will be dimensionless.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of tanh(x)?
A: The tanh function outputs values strictly between -1 and 1 for all real x.

Q2: How is tanh related to exponential functions?
A: tanh(x) can also be expressed as (e^x - e^-x)/(e^x + e^-x).

Q3: What's the difference between tan and tanh?
A: tan is a circular trigonometric function while tanh is a hyperbolic function. Their graphs and properties are different.

Q4: Why is tanh used in neural networks?
A: Its S-shaped curve and differentiability make it useful for backpropagation, though ReLU is now more common.

Q5: What happens when x approaches infinity?
A: tanh(x) approaches 1 as x→∞ and -1 as x→-∞.