Conundrum: Representing Harmonic Waves with Sine Functions

It is claimed that a one dimensional harmonic wave can be represented by the following function

Y = A sin(kx-wt + B)

Where x = location

A = amplitude

K = wave number

w = angular frequency

B = phase adjustment parameter

Here is the conundrum. Can any quantity other than an angle be an argument for a trigonometric function? Let's simplify the problem and ask how can a function such as

Y = sin(x) be interpreted where x is simply a real number representing a geometric position?

In the development of trigonometry we have two methods of defining specific functions. The most rudimentary method is to define them in terms of right triangles.

For example

Sin (theta) = opposite side/ hypotenuse

In this primitive approach right triangles are drawn and it is clear that the argument or preimage of the function is an "angle" and the images of the functions are the ratios of varying sides of the right triangle. In this definitional approach everything relies pretty much on geometric intuitions.

The more sophisticated and general approach is to define the angle q in terms of a unit circle. In this definition the range of the functions become the ratios of coordinates of points on the circumference of the circle divided by the radius of the circle. For example

Sin (theta) = Y/ R

In this approach the angle q is now given a more explicit definition as the ratio of S, an arc length of the circle, divided by the radius of the circle. i.e. theta = S/R

The Y in the above functional definition is the y coordinate of the end point of an arc segment S on a circle of radius r. So literally the trig functions become mappings of ratios. The pre images of the functions are ratios of arc lengths over radial lengths. The images of the functions are ratios of coordinate lengths over radial lengths.

These definitions have physical interpretations as physical units. The angle, the preimage ratio, has units called radians. The image units are simply whatever convenient unit of length we happen to use: meters or yards.

Given the more formal physical definition of angles, the conundrum is this: how do we give physical meaning to sin (x) where x is simply a unit of length from some arbitrary point of origin in some physical reference frame?

In physics, units matter and they matter a lot. Returning to the more complicated harmonic wave function:

Y = A sin(kx-wt + B)

Here we have an even stranger beast showing up as the preimage of the function. In dimensional terms kx would be in meters while "wt" reduces to radians. So the preimage is some kind of mixed unit involving a subtraction of radians from meters!

Now in physics there are many combinations of physical units involving multiplication and division (e.g. v=m/s). But how do we interpret the addition or subtraction of distinct physical units? I'm not sure we can?

Who out there can help me with this conundrum?