I have a rather complex wave function that I'm trying to rotate by $90$ degrees, so that it's drawn from top to bottom on the y-axis instead of left to right on the x-axis. Is there a well-defined way for accomplishing this sort of thing? What if you wanted to rotate your wave by $\theta$ degrees counter-clockwise or clockwise?
Let us assume the wave is defined by the function:
$$y(t) = A\sin(\omega t + \varphi)$$
You talk about the "x-axis" but you've given a function of 't'. I'm going to replace 't' with 'x': $Y = A\sin(\omega X + \varphi)$
A rotation by θ is a change of co-ordinates:
X = sin(θ)y + cos(θ)x
Y = cos(θ)y - sin(θ)x
Applying this to your equation you get:
$\cos(θ)y - \sin(θ)x = A\sin(\omega (\sin(θ)y + \cos(θ)x) + \varphi)$
For the case of rotating 90 degrees specifically, you get:
$x = A\sin(\omega y + \varphi)$
Now, these equations aren't in the form y = f(x). In order to get them into this form, you need to take the equation I've given and solve it for y. In this case, you'll actually find that impossible however. In the general case it's impossible because it's a transcendental equation (which is basically saying it can't be solved because it's the sort of equation that can't be solved).
In the specific case of the 90 degree rotation (and in about half of the other cases as well) it's also impossible because y takes on more than one value for any given x. You can't write y as a function of x if y needs to have multiple values for any given x. One thing you can do though is cut as big a slice of it as possible as follows:
$y = (\arcsin(x/A)-\varphi)/\omega$
Even without solving for y these equations do usefully represent the shape you're describing.