What is the domain of the sine function of a point on a unit circle?

The sine function is an oscillating function, meaning it goes back and forth

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Sine functions go on like that forever. Between (0,2pi) however, the function has a pattern which repeats. As such, we usually only concern ourselves with that part. It just so happens that circles tend to be a useful tool for analyzing this function. The function doesn't actually follow the circle. It does oscillate between 1 and -1 like the circle though. Think of it like this:

You're cranking a manual egg beater. Your hand moves in a circle, but if we were to graph the position of your hand on the y-axis and the time on the x-axis, it would give us a sine function.

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Edit: the domain is all real numbers because the sine function repeats forever in both directions. For every x, there is a value y; thus, the domain is all real numbers. The y values move between 1 and -1 smoothly. Note, that when we graph this, we graph theta (the angle) on the x-axis, not the value of x on the circle (which we put on the y-axis). The reason for this is because the angle is considered to be the dependent variable, while the value is considered to be the independent value (the value of x on the circle depends on the angle)