Why do radians even exist? Why would you use them instead of degrees?
In: Mathematics
The radians are in many ways the “natural” unit for angles. This makes them very convenient especially in modelling the physical world. The arc length of one radian of a circle with a diameter of one feet is one feet (or one metre is one metre). This links angular and linear displacements, velocity and accelerations without annoying conversion factors.
The radians, however, are also a bit of a nuisance for mental math. Which is probably the primary reason we use multiple different units for angles.
A good example of such an intermediate unit is “mil”, which is approximately 0.001 radians. Ten mils at a kilometre equals ten metres.
As a sidenote, “mils” have been so convenient that they were invented several times independently. There are actually three different surviving definitions for “mils”. The most “accurate” splits the circle into 6300 mils (as 2000π ≈ 6283 and a spare milliradians). But, the two more popular ones choose to sacrifice some accuracy for some convenience. Some say 6000 mils to a circle, and others 6400 mils to a circle. The latter is more accurate, but still allows some easy subdivisions. The former is a bit less accurate, but is very well divisible just like 60 minutes are, or 360° are.
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