What is the significance of deciding the convention of $1 \text{ radian} = 180 \text{ degrees}$ over $\pi$? Let's say we alter the definition of the radian, for example 1 radian = 1 degree and there are 360 radians in a circle, then one consequence that I can think of is that the Taylor expansions of trigonometric functions no longer work unless altered accordingly. $$\sin x = x-{x^3\over3!}+{x^5\over5!}-{x^7\over7!}+\cdots $$ 
should instead, in order to work, become
$$\sin x= \left({\pi\over 180}\right)x-\left({\pi\over 180}\right)^3\left({x^3\over3!}\right)+\left({\pi\over 180}\right)^5\left({x^5\over5!}\right)-\cdots$$
My question is: did mathematicians recognize the elegance of defining $1 \operatorname{radian}={180^\circ/\pi}$ in Taylor expansions or also in other stuff (that it simplifies things nicely), or did the said definition come before anything else, or is there other significance to that? It is not important to question an already well-defined structure but I am just curious, thank you.
 A: The big restriction is the differentiation formulas for $\cos(x)$ and $\sin(x)$ only hold when $x$ is measured in radians. The definition of the radian is as follows : 1 radian is defined to be the angle that subtends an arc of length 1 on the unit circle. Using similarity of circles, you get the equation $$s = r \theta$$ where s is the arc-length and $\theta$ is in radians. An arc length of $2 \pi r$ corresponds to the entire circumference of a circle. Plugging in gives$$ 2 \pi r = r \theta \implies 2 \pi = \theta$$ Thus there are $2 \pi $ radians in a circle. Equating $2\pi \text{ rad }= 360^\circ$ gives you the conversion factor between radians and degrees.
So the definition of the radian had absolutely nothing to do with degrees. Defining the radian in the above way gives you the slick formula for derivatives of sinusoids. 
Each definition was independent and it turned out that $1$ radian happened to be the same angle as $\dfrac{360^\circ}{2\pi}= \dfrac{180^\circ}{\pi}.$ 
In one instance you are dividing the circumference of a circle into $360$ evenly-lengthed segments and in the other you divide it into $2 \pi $ segments.
A: From a mathematical point of view you are asking the question the wrong way around.  You seem to be assuming the degree is fundamental and the radian is derived from it to be simpler and get $\frac {180}\pi$s out of the equations.  From a mathematical view the radian is the fundamental unit and the degree is something in the range from a derived unit to a bad mistake.  From an engineering and conceptual point of view degrees are nice because they give simple measures to the parts of a circle we use most (except for my dad who had to cut round desserts into seven pieces)
A: Historically speaking, it is not likely that beautifying the Taylor expansion of sine was “the” reason for defining the radian. After all, the radian was defined by the relationship $r\theta=s$, which is incredibly helpful in physics. Perhaps un-coincidentally, the man who defined the radian in the 1870s, James Thomson, was the brother of the famous physicist Lord Kelvin.
The English mathematician Brook Taylor died almost 150 years beforehand, so I suppose that Taylor expansions could have been considered. In fact, this is one of the reasons we continue to use the radian today. Only when the arguments of sine and cosine are expressed in radians is it true that $\frac{d}{dx}\sin x=\cos x$ and that $\frac{d}{dx}\cos x=-\sin x$, and these relationships are how the Taylor expansion of sine is defined. 
A: As pointed out by others, if you take the unit circle, start at (0,1), and then trace along the circle going clockwise, after you cover a distance of 1 you are at the point where the angle is 1 radian.  It is the natural unit that not only makes sense from that construction, but which supports all the trig functions.  
In fact, a 'radian', in a sense, doesn't exist, but is a convenient name for something which is dimensionless.  Normally physical units (meters, joules, foot-pounds) are inherently definitions of units of some sort and have dimensional units to actually define what '1 joule' is (1 joule equals 1 kg-m per second squared).  The unit (joule) tells you what other measurements and units make it up; there is an entirely different definition with other base units, such as the English system where one unit of work is defined as 1 slug-foot per second squared.
The 'radian' is not a unit at all, but is a more natural measure.  It really goes like this: for any unit circle, the length of the circumference is 2$\pi$ .  That corresponds to one complete angle such that you end up wherever you started.  If you end up only going part way around, the measure of that angle is the length along the circumference.  Thus, a right angle has measure $\pi/2$.  
People throw the term 'radians' there for convenience, but with angles defined this way there are no units; 'radians' are fictitious.  The value falls out of the definition ($s=r\theta$) and the fact that any circle has a circumference of $2\pi r$, and no matter what you measure in (feet, centimeters, light-years), your angle is just a dimensionless number (like $\pi/2$).
A: A degree is a an arbitrary unit used to represent an angle. It is only convention that says 90 degrees is a right angle, 360 degrees is an entire revolution. An equally arbitrary unit, zargs could have say 1000 zargs per revolution.
In contrast, a radian specifies the angle in terms of the circumference enclosed by the angle. It is an actual ratio of circumference traversed vs angle. 
eg. a 90 degree angle encloses 1/4 of the circumference. As the circumference is 2pi*r then the ratio is 2*pi*r/(r * 1/4) or pi/2. On this basis 90 degrees = pi/2.
So in actual fact radians are a real way of representing angles, degrees are an arbitrary construct defined as 360 degrees = 1 revolution. 
