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sinh x is half the difference of e x and e −x cosh x is the average of e x and e −x In terms of the exponential function : [ 1 ] [ 4 ] Hyperbolic sine: the odd part of the exponential function, that is, sinh x = e x − e − x 2 = e 2 x − 1 2 e x = 1 − e − 2 x 2 e − x . {\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e ...
For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of § Definitions in terms of logarithms do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected.
Since cosh x + sinh x = e x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers n, ( + ) = + . If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x) n. [4]
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A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
For example, (+ /) converges to the exponential function , and the infinite sum = ()! turns out to equal the hyperbolic cosine function . In fact, it is impossible to define any transcendental function in terms of algebraic functions without using some such "limiting procedure" (integrals, sequential limits, and infinite sums are just a few).
The analog of the Pythagorean trigonometric identity holds: [2] + = If X is a diagonal matrix, sin X and cos X are also diagonal matrices with (sin X) nn = sin(X nn) and (cos X) nn = cos(X nn), that is, they can be calculated by simply taking the sines or cosines of the matrices's diagonal components.
In the integral , we may use = , = , = . Then, = = () = = = + = +. The above step requires that > and > We can choose to be the principal root of , and impose the restriction / < < / by using the inverse sine function.