The hyperbolic sine, denoted as sinh, is a mathematical function that describes the relationship between the angles of a hyperbola, similar to how the regular sine function relates to a circle. It is defined as the average of the exponential function: sinh(x) = (e^x - e^(-x)) / 2, where e is the base of natural logarithms. This function is useful in various fields, including physics and engineering, particularly in calculations involving hyperbolic geometry. The hyperbolic sine function has properties similar to the regular sine function, such as being an odd function, meaning sinh(-x) = -sinh(x). Its graph resembles that of the regular sine function but grows exponentially as x increases. The hyperbolic sine is often used in solving differential equations and modeling phenomena like wave motion and heat transfer.