I feel that a radian measurement is a real number because we use it widely in calculus and the evalu

1. ${180}^{\circ }, \pi \text{ rad} , 7 \text{ cm}$ are all physical quantities, each having both a numerical value and a unit.
An angle can be thought of as a ratio of lengths; it has a degree count of
$\frac{180}{\pi }\times \frac{\text{length of the arcthat subtends the angle at a circle's centre}}{\text{radius of thecircle}}.$
Thus, an angle is a dimensionless quantity; so, its SI derived unit is just 1. (On the other hand, distances have SI unit metre.)
Just to be clear, an angle is measured or defined in radians, degrees, gradians, etc.

2. 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 rad having been divided out so that the domain really is $\mathbb{R}$), and both returning the same output for equivalent inputs. (To distinguish between them, some authors call the latter the natural trigonometric functions.)
2.1. $\sin ( \pi ) \ne \sin ( 180 ) = \sin ( {10313}^{\circ }) .$
2.2. The Taylor series ${\displaystyle \sin ( x ) = x - \frac{x^3}{3 ! }+ \frac{x^5}{5 ! }- \frac{x^7}{7 ! }+ \dots }$wouldn't be consistent if x carries any unit.
2.3. The only sine that can be recursively compounded is the unitless sine.
$\sin ( \sin \left( \frac{\pi }{4}\right) ) = \sin ( \frac{{180}^{\circ }}{\pi }\sin \left( {45}^{\circ }\right) ) ,$
whereas
$\sin ( \sin ( {45}^{\circ }) )$
is not meaningful.
2.4. ${\displaystyle \frac{\mathrm{d} }{\mathrm{d} x }}\sin ( x^{\circ }) = \frac{\pi }{180}\cos ( x^{\circ }) ;$
the derivative of $\sin ( x^{\circ })$ at $x = 60$is ${\displaystyle \frac{\pi }{360}, }$not ${\displaystyle \frac{1}{2}. }$

3. Similarly, in the arc length formula $s = r \theta ,$ the subtended angle is $\theta \text{ rad} ,$ not $\theta .$
3.1 $“ s = r ( \pi ) ”$ and $“ s = r ( \pi \text{ rad} ) ”$ are not synonymous; the latter is as incoherent as $“ s = r ( {180}^{\circ }) ” .$

4. The aforementioned examples show that radian is the natural angular unit, as opposed to degree and gradian. In fact, the unit "rad" is typically omitted in mathematics where the context is sufficient.