Is 1 rad important?

Question

Of course radians generally come in ratios of π.

So is 1 rad important/useful/special? Or, for that matter, is any integer radian measure important?

Besides being approximately 57°, I can't seem to find anything.

As an alternative way of asking this question — are the values of $sin(1)$, $cos(1)$, $tan(1)$, etc important (I know they are transcendental)?

Answer 1

Radians do not come in ratios of π, we just call a half rotation π radians, it is the natural thing to do. Degrees are, in my opinion, nonsense. What is the significance of the number 180? It is very clear that π has a very real connection to angles - πr is the length of the arc of a semicircle radius r. It is natural to associate π with this type of angle, then.

Are integer radians very "important?" I don't know what that means, nor do I know why they should be important.

Radians are a very natural angle measure. Many formulas become simplified, and even simple things that we take for granted (d/dx sin(x) = cos(x)) are only true for an argument in radians. It is because the relationship between angles and the number π is very concrete.

Answer 2

Radians are important because when you measure angles in radians you get neat formulae for the derivatives of trigonometric functions, for power series, and you have things like the arc length for an angle $\theta$ is $r\theta$ and formulae without unnecessary factors or constants involving angular velocity and angular momentum.

In short, in more advanced work, radians turn out to be the "natural" measure, rather than the arbitrary degrees. This is similar to the way in which $e$ is the natural base for logarithms, except when trying to do calculations in base $10$.