Is a degree measurement a real number? I feel that a radian measurement is a real number because we use it widely in calculus and the evaluation of limits of real and complex functions; but I've never heard, for example, someone consider $$\lim_{x\to 0} \frac{\sin x}{x}=1$$ when $x$ is in degrees.
Is an angle in degrees not a mathematical quantity, but a physical quantity?
Is a degree measurement a real number?
 A: *

*$180^\circ,\, π\text{ rad},\, 7\text{ cm}$ are all physical
quantities, each having both a numerical value and a unit.
An angle can be construed as a ratio of lengths: its number
of degrees is $$\frac{180}\pi\times\frac{\text{length of the arc
that subtends the angle at a circle's centre}}{\text{radius of the
circle}}.$$ Being a measure of some quotient of lengths $\left(\frac{\textrm m}{\textrm m}=1\right),$ an angle is a dimensionless quantity. To be clear though, an angle does have units: it is measured/specified in radians, degrees, gradians, etc.


*It's instructive to understand the two standard versions of each
trigonometric function as being in fact two different
functions: one accepts an input with unit $^{\circ}$, the other
accepts a unitless input (the $\textrm{rad}$ having been divided out
so that the domain really is $\mathbb{R}$ or $\mathbb{C};$ each element of the former corresponds to but isn't an angle), and both returning the
same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)

*

*$\sin(\pi)\neq\sin(180)=\sin(233^{\circ}).$

*The Taylor series $\displaystyle\sin(x)=x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots$ wouldn't be consistent if $x$ carries any unit.

*Only the unitless version of sine can be recursively composed $$\sin\left(\sin\left(\frac{\pi}{4}\right)\right) = \sin\left(\frac{180^{\circ}}{\pi}\sin\left(45^{\circ}\right)\right),$$ whereas $$\sin\left(\sin(45^{\circ})\right)$$ is not meaningful.

*$$\dfrac{\mathrm{d}}{\mathrm{d}x}\sin(x^{\circ})=\frac{\pi}{180}\cos(x^{\circ});$$ the derivative of $\sin(x^{\circ})$ at $x{=}60\:$ is $\displaystyle\frac{\pi}{360},$ not $\displaystyle\frac12.$



*Similarly, in the arc length formula $s=r\theta,$ the subtended
angle is $\theta\textrm{ rad},$ not $\theta.$

*

*$“s=r(\pi)”$ and $“s=r(\pi\text{ rad})”$ are not synonymous; the latter is as incoherent as $“s=r(180^{\circ})”.$



*The above points illustrate that unlike degree and gradian, radian is the natural angular measure. So much so that in mathematics, the unit "$\textrm{rad}$" is generally dropped whenever the context is sufficient.
A: Obviously, a unit of measurement (which doesn't measure quantity) can't be a real number. 
However, the measurements with that unit can be a real number (2$\pi$ radians, 54 degrees, 0.23 revolutions, ...)

In your question, you have stated that "radian" should be a real number because we use it widely in calculus. However, the example you gave ($\lim_{x \to 0} \frac{\sin x}{x} $) doesn't imply that a radian is a real number (specifically because sin x is a real number, and sin is defined in radians). You might find other equations in calculus or trig that involve radians, but they always fall into one of these categories:

*
    
*those equations directly use radians (an oversimplified example can be $\frac{d\theta}{dx}$), which returns an answer in (an unit used to measure angles). There aren't any connections between the radian (or any other units) and real numbers here.
    
* those equations involve radians, but actually returns real numbers. In that case, there must be a function that takes in an angle measurement and returns a real number (take the trig functions for example) (you won't find any equations that doesn't involve those functions). Which, again, doesn't make any connections between radians and real numbers.

Degrees are also an unit that we use to measure angles (1 degree = $\frac{\pi}{180}$ radians). Therefore, we can have valid equations (not just in calculus but in other fields as well) using degrees.
we just need to multiply the measurement in degrees with $\frac{180}{\pi}$
In conclusion, radians (and degrees) are just units that we use to measure angles (in the same way that a meter is used to measure lengths or a gram is used to measure mass), and they aren't equivalent to any real numbers (in the same way that you can't convert a meter to kilograms).
(sorry if this answer is confusing - this is my first answer ever)
