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The hyperbolic sine function, \sinh x, is one-to-one, and therefore has a well-defined inverse, \sinh^{-1} x, shown in blue in the figure. In order to invert the hyperbolic cosine function, however, we need (as with square root) to restrict its domain.
If x = sinh y, then y = sinh-1 a is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigonometric functions we restrict ourselves to principal values for which they can be considered as single-valued.
5 Απρ 2023 · Here are the steps to prove that sinh − 1(x) = ln(x + √x2 + 1) using the given equations: Start with the expression x = ey − e − y 2, which relates x and y through the inverse hyperbolic sine function sinh − 1(x). Substitute this expression for x into the right-hand side of the equation to be proven: ln(x + √x2 + 1).
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All you have to do is input the value of one of the functions (for example, sinh x \sinh x sinh x or tanh x \tanh x tanh x), and this tool will automatically return the value of x x x. The formulas used to compute inverse hyperbolic functions are shown below.
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