Why are turns not used as the default angle measure?

Solution 1:

Because if you use $1$ for the turn instead of $2\pi$ (forgetting what a radian is), you get that $\frac{\mathrm d}{\mathrm dx} \sin x = 2\pi\cos x$, which is probably not what you want, and many other problems, such as $\sin x$ not being a solution to $y''+y=0$ etc.

You can discuss whether the full turn is $2\pi$ or $\tau$ (tau), but you can't change the fact that it's equal to $6.2831853071\cdots$.

Solution 2:

There is a movement to make $2\,\pi=\tau$ a fundamental constant instead of $\pi$ (read The Tau Manifesto.) But not as far as I know to use $1$ as the measure of the whole circunference. One possible reason is that $\pi$ encodes the relation between the radius an the length of a circumference: a circular sector of radius $r$ and angle $\alpha$ radians has a length equal to $\alpha\,r$; it would be $2\,\beta\,\pi\,r$, where $\beta$ is the measure of the angle in the units you propose. Moreover, you cannot avoid $\pi$ in formulas like $$ e^{\pi i}+1=0. $$

Solution 3:

In fact, "radians" are not a unit like meters or second, so you can't rescale them to make $2\pi$ become $1$ (as sometimes you do in physics, rescaling e.g. meters to make $c=1$).

As an aside, degrees are an absolutely artificial concept, and you should try to never use them.

Solution 4:

Radians are the natural, dimensionless choice of unit to measure angle. We could certainly define the full turn as $1$. That's a nice, wholesome unit. It's nicer than pushing our historical comfort with base $60$ with $360^\circ$ or dealing with cumbersome $400 \text{ gon}$. But does $1$ really reflect the system we're trying to describe?

In choosing an appropriate unit, we should ask "What is an angle measure?" One appropriate definition is the measure $\theta$ of the angle subtended by an arc length $s$ of a circle of radius $r$. This definition, though, needs some work to become a practical tool. Given the obvious relationship between an angle and a circle, we can call upon Euclidean geometry to find ourselves a nice parameter. Behold the ratio of circumference to diameter: $C/d=\pi$! Now we observe that the relationship between arc length and circumference goes as

$$s = (\text{some fraction})\times C = \dfrac{\theta \times C}{\theta_{turn}}$$

We're getting somewhere. Let's adopt a convenient parametrization of angle measure, something we can throw numbers at and interpret easily. Perhaps a convenient choice would necessitate a convenient circle. Well, the unit circle has a nice radius and area. Its circumference is $2\pi$. If the distance around the unit circle is $2\pi$, then why don't I adopt this for my unit of angle measure? I'll define the angle measure $\theta_{turn}$ that takes me around the circle to be $2\pi$.

Arc length is now cleanly given by

$$s = r\theta $$

That's the definition of radian in that last line. Why is it a superior choice? Any other choice of units would have left over some other constants. With the choice of $2\pi$, we have a clean, relevant, and dimensionless unit.