What is the advantage of measuring an angle in radian(s)? What is the advantage and use of measuring an angle is radian(s) compared to degree(s)? My book suddenly switched to radian(s) for measuring an angle in this grade and I do not know why.
Look at the following picture. It is a circle of radius $r=1$ and there is an angle $\alpha$ that cuts out an arc $c$ from the circumference of the circle.

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

How long is this arc? Well, this is where radians are immensely beneficial. If you express the angle in radians, then the length of the arc is exactly $\alpha$. If $\alpha=0.123$, then the arcs length is $0.123$ too. Easy.

You know that the full angle in radians is $2\pi$. If you choose $\alpha=2\pi$ in the above image, you would describe an arc $c$ that is actually just the full circle. And you immediately see, that its circumference (here also the length of the arc) is $2\pi$ too. This is exactly what the formula for the circumference $U(r)=2\pi r$ would give you anyway.

Of course this is also useful for circles of radii other than $1$. If your circle has a radius $r$ and you look at an arc cut out by an angle $\alpha$ in radians, then the length of the arc is $\alpha r$. It just scales linearly.



This is the visual aspect. But there are more (mathematical) reasons. Ask yourself, what would be a natural unit to measure an angle in? Degree is not very natural. There is nothing special about the number $360^°$ but its high divisibility. You could measure an angle in the interval $[0,1]$. Or you can use its natural connection to the circle that I explained above.

However, there are many mathematical functions that take angles as inputs, e.g., $\sin(x),\cos(x)$, etc. But have you ever asked how to actually compute $\cos(x)$ without a calculator? There are formulas that give very good approximations, e.g.,

$$\cos(x)\approx 1-\frac12 x^2+\frac1{24}x^4.$$

But they only work for radians. You can write them down for other units, but they will never look so natural, not even for angles in the units $[0,1]$.
There is no "advantage" of radians over angles, for example. Think of it as an unit of measure. For example, kg vs pounds. The only difference is on the choice one makes, and not in what actually represents.  

EDIT:

Do not be misled by the down votes. This is an answer from the perspective of a theoretical physicist.

I strongly encourage you to think in, again, units in physics. There is no advantage in using one units or the other, except perhaps your final numerical results and the application of what you want to give them. The theory does not change at all if you use degrees or radians, the angle is still the same angle viewed with different glasses! I have never used degrees since I started the career (and not from that I am going to say that one is advantageous one over the other, because I can google up examples where there is a wide application of degrees over radians), because I don't need them. Perhaps and engineer would need them more and that would make them advantageous for him.

I want to stress that the theory is not change by either using degrees or radians, and from this point of view the advantage type of feeling you have is not really relevant.

If you continue your studies in science, you will find specially in Physics, that most of the equations are given in free-unit form. Maxwell's equations describe the same phenomena no matter which units you choose (the only difference are a couple of prefactors in the equations, which will give you your chosen units). Newton's equations are still the same, whether you choose to measure forces or masses in  cgi or international unit system (or any other system unit).