Why do we use radians for polar coordinates rather than degrees? Is there a specific reason why radians must be used when converting from Cartesian to Polar coordinates, is it just a convention or does it not matter at all?
 A: The radian has a geometric meaning, it is the ratio between the arc's length and the circunference's radius.
Other measures of angle have arbitrary scales, so numerical values in degrees or gradians have no special meaning.
Maybe because of their more 'natural' definition, by using radians, we find ellegant derivatives and power series for the trigonometric functions (exponentials too, because of Euler's formula).
A: If I were to ask you "What is a degree?", could you say anything other than "There are 360 degrees in one revolution"? And then I would ask you "Why 360? Why not 100? Why not 17? Is there anything natural about 360?"
On the other hand, radians are a natural measure of an angle. Given an angle, form a circle (of any radius you like) centered at the vertex of the angle. The radian measure of the angle is exactly the number of times the radius length fits on the arc of the angle. It's independent of whatever radius you chose.
When angles are measured in radians, you also have nice relationships between trig functions like $$(\sin \theta)' = \cos\theta$$
$$(\sin \theta)'' = -\sin \theta$$
A: Radians provide a nice association between the area of a section of a circle and the size of the angle.  i.e. $A = \frac 12 r^2 \theta.$  They also behave nicely with arc lengths.  $C = r\theta$
And so if you have something like the rotating wheel, and you want to know the distance a belt attached to that wheel has moved, then rotation in radians avoids a conversion factor.   This makes radians handy for problems in physics and engineering.
But the bigger bonus of this association between arc length and radians comes up when $\theta$ is small.  For small $\theta, \sin\theta \approx \theta.$  And that is really important when you get to calculus.  Once you start calculus you will wonder why you ever thought degrees were easier.
We haven't really talked about polar coordinates.  And, honestly, there is no reason you cannot use degrees when you work with polar coordinates.  And plenty of "real-world" applications use degrees and polar coordinates.  Problems in navigation, for example, are almost always done in degrees, and are effectively problems in polar coordinates.
Your teachers want you to use radians to be more comfortable with them when the time comes that degrees become inefficient.
A: *

*It makes computation of circular arc lengths easier: no conversion factor like $\pi/180$ needed.

*It makes the Taylor series for functions like $\sin$ and $\exp$ look cleaner.
But yes, it's a convention.
A: The main reason is that the derivatives of sinusoidal functions only work if the angles are measured in radians.
A: As many others commented, radians are the natural unit.
Degrees exist only for historical reasons and are not an extremely practical unit (not mentioning the minutes/seconds issue). One of their only assets is that $360$ is divisible by $2,3,4,5,6,8,9,10$ and others.

IMO, we are lacking a truly practical unit, which does not involve the constant $2\pi$: revolutions, i.e. a full turn corresponding to $1$ unit.
