While dealing with trigonometric functions, we usually assume the argument to be in radians. The operator then returns a unitless output. Ie: the range of sin x for instance would be a set of certain ratios.
By the same logic, the argument of sine in sin(cos(x)) must also be in radians. However, cos x will always be a unitless quantity ( being a ratio ). How then, are composite trigonometric functions, to be evaluated?
Great question.
Answer: Radians are actually unitless. In fact, one way to define an angle is to say "it's a (technically nice enough) subset of a circle." We then say that the measure of the angle (which is the kind of thing that we allow as an argument to sine or cosine) is the arclength of the subset divided by the radius of the circle. That's a quotient of two distances, so it's unitless.
The peculiar thing here is that we're very sloppy in naming an angle (in classical geometry: a pair of rays with the same origin; in my definition above, a subset of a circle), and in naming the "measure" of that angle (which is a real number). We tend to use the name "$\theta" for both of these things, which can confound stuff until you get used to it.
Still, one might ask "if radians are unitless, why do we use them at all?" I don't really have a good answer here, except that people also measure angles in degrees (which are also unitless!), and it's nice to have a way to say "I'm converting from degrees to radians", when in fact what we're doing, when we write
$$ \sin x $$
and $x$ is "an angle in degrees", is that we're really computing
$$ sind(x) $$ where $sind$ is an entirely distinct function from $\sin$, and can be defined by $$ sind(x) = \sin (\frac{\pi x}{180}). $$
You can tell that $sind$ and $\sin$ are different functions, by looking at their values on the real number 180. The value of the first on $180$ is zero; the value of the second is about $-0.8$.
Still, people find the idea of two different functions more baffling than that of two different units, so we pretend radians and degrees are "units" of angle measure.
The real $\sin$ function is defined as a function that takes a real number as an input and produces a real number as an output. It does not take an angle as an input. (What is an angle anyway?) It just so happens that one can interpret the relation between input and output of $\sin$ in geometric terms. (If you move anti-clockwise around the unit circle in the Cartesian plane starting from the point $(1,0)$ for a distance of $t$, then your $x$-coordinate would be $\sin(t)$. But what is distance along an arc anyway?)
Now in the past we used degrees to measure angles, rather than the distance along the unit circle arc. Still, there is only one $\boldsymbol{\sin}$ function. We simply define $^\circ = \frac{2π}{360}$. Then $\sin(30^\circ) = \sin(\frac{π}{6}) = \frac12$, for example.
