Can the sinh and cosh functions be expressed in terms of each other?

[Jhenrique]

Is possible express the sine in terms of cosine and vice-versa:

$\sin(x) = \cos(x-\frac{\pi}{2})$

$\cos(x) = \sin(x+\frac{\pi}{2})$

So, of some way, is possible express the sinh in terms of cosh too and vice-versa?

 

[lurflurf]

sure

$\sinh(x)=-\imath \, \cosh(x+\pi\,\imath/2) \\ \cosh(x)=-\imath \, \sinh(x+\pi\,\imath/2)$

 

[Jhenrique]

sure

$\sinh(x)=-\imath \, \cosh(x+\pi\,\imath/2) \\ \cosh(x)=-\imath \, \sinh(x+\pi\,\imath/2)$

No exist an identity that no involve imaginary unit?

 

[HallsofIvy]

No, because that is the whole point of the hyperbolic functions: sinh(x)= i sin(ix) and cosh(x)= cos(ix).

 

[Curious3141]

Of course, there's the 'obvious' one which you can find by rearranging $\cosh^2 x - \sinh^2 x = 1$, but I don't think that's what you're going for, is it?

 

[Jhenrique]

Of course, there's the 'obvious' one which you can find by rearranging $\cosh^2 x - \sinh^2 x = 1$, but I don't think that's what you're going for, is it?

no, but thanks

 

[lurflurf]

I don't know if this is what you want either, but I forgot to mention http://en.wikipedia.org/wiki/Gudermannian_function]Gudermannian[/PLAIN] function.

$$\mathrm{gd}(x)=2\arctan[\tanh(\tfrac{1}{2}x)]=2\arctan(e^x)-\tfrac{1}{2}\pi$$

 

[docarlson]

Considering the graphs of sinh(x) and cosh(x), they don't have the shift property that sin(x) and cos(x) have for x in set of real numbers.

 

[mathman]

cosh²(x) - sinh²(x) = 1.