The hyperbolic tangent function, commonly known as tanh, is a mathematical function that describes the relationship between the hyperbolic sine and hyperbolic cosine functions. It is defined as the ratio of these two functions: tanh(x) = sinh(x) / cosh(x). The output of tanh ranges from -1 to 1, making it useful in various applications, particularly in neural networks. In addition to its mathematical significance, tanh is often used in machine learning as an activation function. It helps to introduce non-linearity into models, allowing them to learn complex patterns. The smooth gradient of tanh aids in optimizing algorithms, making it a popular choice for training deep learning models.