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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.
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Omni's hyperbolic sine calculator is very straightforward to use: just enter the argument x, and the value of sinh(x) will appear immediately! And you can calculate inverse sinh as well! Just fill in the field sinh(x), and the value that appears as x is exactly the value of the inverse function. And this is not all!
In the points z=2 pnä’m’;n˛Zìm˛Z, the values of the hyperbolic functions are algebraic. In several cases, they can even be rational numbers, 1, or ä (e.g. sinhHpä’2L=ä, sechH0L=1, or coshHpä’3L=1’2). They can be expressed using only square roots if n˛Z and m is a product of a power of 2 and distinct Fermat primes {3, 5, 17 ...
It is a tool that computes the values of six basic hyperbolic functions – sinh, cosh, tanh, coth, sech, and csch – all in a blink of an eye. You can also use it to calculate the inverse hyperbolic functions.
Prove the formula for the derivative of y = sinh −1 (x) y = sinh −1 (x) by differentiating x = sinh (y). x = sinh (y). (Hint: Use hyperbolic trigonometric identities.)
¶nShiHzL ¶zn −dnShiHzL-â k=0 n-1äkH-1LnHn-1L!zk-n k! sinh äpk 2 +z’;n˛N 06.39.20.0003.01 ¶nShiHzL ¶zn −dn ShiHzL-Boolen„0,Hn-1L!â k=0 n-1H-1LnH-äLkzk-n k! sinhz-äkp 2 ’;n˛N 06.39.20.0004.02 ¶nShiHzL ¶zn −2n-2pz1-n 2F ” 3 1 2,1; 3 2,1-n 2, 3-n 2; z2 4 ’;n˛N Fractional integro-differentiation 06.39.20.0005.01 ...
