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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, = = =.
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.
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.
The exponential function e x for real values of x may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of e z for complex values of z simply by substituting z in place of x and using the complex algebraic operations. In ...
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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Then exchange all the cosine and sine terms to cosh and sinh terms. However, for all products or implied products of two sine terms replace it with the negative product of two sinh terms. This is because − i sin ( i x ) {\displaystyle -i\sin(ix)} is equivalent to sinh ( x ) {\displaystyle \sinh(x)} , so when multiplied to together the ...