Does the relation $\sinh(iz) = i\sin(z)$ have anything to do with a rotation of the complex plane? Ok, I recently learned about the following relation in complex analysis:
$$i\sin(z) = \sinh(iz)$$
Now, let $\sin(z)$ be the image $I_1$ of the complex plane $\mathbb{C}$, and $\sinh(iz)$ be another image $I_2$ of the complex plane $\mathbb{C}$. Since the images are sets of vectors if we think of the complex plane as $\mathbb{R}^2$, then since the relation above holds, does this mean that every vector in $I_2$ is rotated by $\pi/2$ radians CCW in relation to the corresponding vector in $I_1$. So, in essence, does this mean that if we just decided to rotate the second image on our complex plane by $\pi/2$ radians, we would get the first image?
How can one understand this geometrically, if that's the case so to speak. What does really happen here, and is there any visualization behind this.
Thanks.
 A: As you should know, $\cos$ and $\sin$ are sometimes called “circular functions”, because the curve formed by $(x = \cos(t), y = \sin(t))$ is a circle, whose equation can also be written as $x^2 + y^2 = 1$.  Similarly, $\cosh$ and $\sinh$ are called “hyperbolic functions”, because $(x = \cosh(t), y = \sinh(t))$ is (half of) a hyperbola, whose equation can be written as $x^2 - y^2 = 1$.
Although the two curves look very different when you zoom out, they're the same at $t=0$: $(1, 0)$.  Positive $t$ brings you up into Quadrant 1, and negative $t$ brings you down into Quadrant 4.  And the tangent lines to both curves are vertical there.
So, it doesn't really make sense to speak of a “rotation” between circular and hyperbolic functions, since both are defined relative to the positive x-axis.  The $i$ factor means something else.  Let's take a closer look at the hyperbola's x-y implicit equation.
$$x^2 - y^2 = 1$$
$$x^2 + (-1)y^2 = 1$$
$$x^2 + (iy)^2 = 1$$
This looks a lot like the equation of an ellipse.  In a way, you could say that a hyperbola is an ellipse with an imaginary aspect ratio.  And that's where the $i$ comes in.  Not rotation, but scaling.
