Does $e^{i\theta}$ Relate To Hyperbolic Sine/Cosine?

Question

I would like to understand the relationship betwene $e^{i\cdot \theta}$ and hyperbolic sine and cosine. Here is what I have done so far:

Given: $$\sinh(x)+\cosh(x)=e^x $$ $$i\sin(\theta)+\cos(\theta)=e^{i\theta} $$

Replaced $x$ in the first equation with $i\theta$ :

$$\sinh(i\theta)+\cosh({i\theta})=e^{i\theta} $$

Given: $$i\sinh(x)= \sin(ix)$$ $$\cosh(x)=\cos(ix) $$ Replaced $x$ with $i\theta$ $$i\sinh({i\theta})= \sin(i\cdot{i\theta})=\sin(-\theta)$$ $$\cosh({i\theta})=\cos(i\cdot {i\theta})=\cos(-\theta) $$

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

I feel like I'm doing meaningless symbolic manipulation that likely is flawed on some level.

It only feels meaningless until you realise that all these functions are defined on $\mathbb{C}$, and not only on $\mathbb{R}$. Then you understand that these are fundamental identities. — Daniel Fischer, Oct 16 '13 at 18:30
Now replace $x$ with $-i \theta$ and add the resulting equations. Don't forget that sine is odd and cosine is even! — The Chaz 2.0, Oct 16 '13 at 18:34
http://math.stackexchange.com/questions/455480/how-were-hyperbolic-functions-derived-discovered — User3910, Oct 16 '13 at 19:48

score 2 · Answer 1 · answered Oct 16 '13 at 18:33

2

You could have obtained these identites a little easier using the definitions $$\begin{align*} \sin(x) & = \frac 1 {2i} (e^{ix} - e^{-ix}) \\ \cos(x) & = \frac 1 2 (e^{ix} + e^{-ix}) \\ \sinh(x) & = \frac 1 2 (e^x - e^{-x}) \\ \cosh(x) & = \frac 1 2 (e^x + e^{-x}) \end{align*}$$

As @DanielFischer said, these make much more sense when considered as whole functions on $\mathbb C$.

answered Oct 16 '13 at 18:33

AlexR

24,905

Does $x=\theta$ here? – User3910 Oct 16 '13 at 18:37
@JoeHobbit $x\in\mathbb C$ is just any variable. It can even be $\text{car}$, but noone wants to write $$\sin(\text{car})$$ ;-) – AlexR Oct 16 '13 at 18:38
@AlexR That's indeed not a good name for a variable. Now $\sin (\text{bad})$, that makes sense. – Daniel Fischer Oct 16 '13 at 18:42
@DanielFischer and $\sin(\text{good}) \approx \sin(\text{food})$? – AlexR Oct 16 '13 at 18:43
It seems you missed the pun. You're not a seafarer, are you? – Daniel Fischer Oct 16 '13 at 18:45
@DanielFischer Nope, neither mother tongue - it's easy to miss those puns then I guess :) – AlexR Oct 16 '13 at 18:47

Does $e^{i\theta}$ Relate To Hyperbolic Sine/Cosine?

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