Difference between revisions of "Hyperbolic trig functions"

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

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.

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