Difference between revisions of "Hyperbolic trig functions"

Latest revision as of 20:00, 3 November 2024

The Hyperbolic trig functions can be thought of the classical trig functions except found on an unit hyperbola. There are as follows:

$\sinh(x)=\frac{e^x+e^{-x}}{2}$

$\cosh(x)=\frac{e^x-e^{-x}}{2}$

$\tanh(x)= \frac{\sinh{x}}{\cosh{x}} =\frac{e^x+e^{-x}}{e^x-e^{-x}}$

They behave much like normal trig functions, as most of the identities still hold. They do so because:

$\sinh(x)= -i\sin{ix}$

$\cosh(x)=\cos{iz}$

$\tanh(x)= -1\tan{iz}$

However, the arc-hyperbolic trig functions have different derivatives.

This article is a stub. Help us out by expanding it.

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 The Hyperbolic trig functions can be thought of the classical trig functions except found on an unit hyperbola. There are as follows:
-<math>\sinh{x}=\frac{e^x+e^{-x}}{2}</math>
-<math>\cosh{x}=\frac{e^x-e^{-x}}{2}</math>
+<math>\sinh(x)=\frac{e^x+e^{-x}}{2}</math>
-<math>\tanh{x}= \frac{\sinh{x}}{\cosh{x}} =\frac{e^x+e^{-x}}{e^x-e^{-x}}</math>
+<math>\cosh(x)=\frac{e^x-e^{-x}}{2}</math>
-Also:
+<math>\tanh(x)= \frac{\sinh{x}}{\cosh{x}} =\frac{e^x+e^{-x}}{e^x-e^{-x}}</math>
-<math>\sinh{x}= -i\sin{ix}
+They behave much like normal trig functions, as most of the identities still hold.
+They do so because:
-</math>\cosh{x}=\cos{iz}
+<math>\sinh(x)= -i\sin{ix}</math>
-<math>\tanh{x}= -1\tan{iz}</math>
+<math>\cosh(x)=\cos{iz}</math>
+<math>\tanh(x)= -1\tan{iz}</math>
+However, the arc-hyperbolic trig functions have different derivatives.
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