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For various reasons already discussed in other stackexchange posts, we implicitly use radians in trigonometric functions by convention. For example, one period of $sin(x)$ lies in $0 \leq x < 2\pi$, when we could have instead used $\sin_{\text{degree}}(d)=\sin(\frac{\pi}{180}d)$ or $\sin_{\text{cycle}}(c)=\sin(2 \pi c)$ where one period lies in $0 \leq d < 360$ or $0 \leq c < 1$.

So how did we still end up using degrees and cycles? Angles are measured in degrees more often than radians in basic geometry content, where $\sin$ may even mean $\sin_{\text{degree}}$ with no warning. Ordinary frequency (cycles per second a.k.a hertz) is often used instead of angular frequency (radians per second).

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    $\begingroup$ Why do we still use hours, minutes, feet, yards, miles, etc., when we have the decimal system and SI units? Because it is easier to occasionally convert than to undertake a massive transformation of existing practice with the accumulated corpus of habits, intuitions, laws, standards, documents, etc. This is called social inertia. $\endgroup$
    – Conifold
    Apr 1, 2021 at 0:47
  • $\begingroup$ I would love a small expansion on the (relatively recent) history of the conventions. Did everybody use degrees first, then someone thought radians looked nice? Why didn't we all convert to a standard? Is it tied to nationality like the use of imperial units? Who brought cycle/hertz into the mix? $\endgroup$
    – BatWannaBe
    Apr 1, 2021 at 1:04

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We do not use radians "by convention". Radian is a necessary intrinsic measure of an angle, which is related to the "fact of nature" that the length of a unit circle is $2\pi$. Turns is also a natural measure, which is sometimes preferred to radians, when it is convenient.

Degrees are different, the reason of using them is purely historical, related to the ancient Babylonian numeration system, with base 60. So the use of degrees, and other similar things is motivated by history tradition and convenience. Like hours, grads and other measures of angles of historic/cultural origin.

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    $\begingroup$ Well yes it's not purely convention, there are very good reasons for choosing one measure or another. In that same vein, I could also say it was a good idea for many ancient cultures developing astronomy to use a highly composite number like 360 that was close to the number of days in a year. By convention I just mean that we found a few "natural" ways to measure angles and we choose one in some contexts. I'm looking for a more historical perspective, like how and when it was decided that software implementations of $sin$ should use radians; that would certainly have a big impact. $\endgroup$
    – BatWannaBe
    Apr 1, 2021 at 1:26
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    $\begingroup$ Good software lets you choose how to measure angles. Even my 25 year old Casio calculator permits me to switch between degrees and radians. $\endgroup$ Apr 1, 2021 at 2:04
  • $\begingroup$ Used to have one like that, good times. Now I'm typing deg2rad everywhere. $\endgroup$
    – BatWannaBe
    Apr 1, 2021 at 4:07
  • $\begingroup$ @BatWannaBe if you are really "typing deg2rad everywhere," you should write your own wrapper function "sind, cosd" etc. which take degrees instead of radians as input. Several languages (MATLAB) have these built in. $\endgroup$ Apr 1, 2021 at 11:00
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    $\begingroup$ @Carl Witthoft: "radians transcendental"?? They can be rational, or integers:-) $\endgroup$ Apr 1, 2021 at 11:04
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I ran across an answer to my satisfaction, but it sort of turns the question on its head: "Why do we implicitly use radians in trigonometric functions if we still use degrees and cycles?" It's not as historical as I imagined, but there is a bit.

Each choice may arise more naturally in different contexts:

  1. Degrees: integers for a wide variety of angles, early astronomy (days in year), chords geometry
  2. Cycles: rotations, describing periodic signals as periods per unit time
  3. Radians: arc geometry, limits scaling, derivatives scaling

Each choice lets you do the same math, but once we hit the Computer Age, the radian's advantage with limits became indispensible.

I'm not going to describe the entirety of the iterative CORDIC algorithm or its applications, just an overview for sine and cosine. With each iteration $k$, it adjusts the angle of a unit vector by $\arctan (2^{-k})$ toward the target angle, so that the unit vector's position (sine and cosine) can be adjusted with cheap bit-shifts instead of generic multiplication. $\arctan$ isn't cheap, of course, but $\arctan (2^{-k})$ can be precomputed for a natural number sequence $k$ and stored in a lookup table. It is the lookup table where the choice of degree/cycle/radian is made.

With radians, $\lim_{x \to 0^+} \arctan (x)/x = 1$, so at large enough $k$, you can take $\arctan (2^{-k}) = 2^{-k}$ and don't need to precompute further. Adjustment by $2^{-k}$ only requires changing a single bit. By contrast, $\lim_{x \to 0^+} \arctan_{cycle} (x)/x = 1 / (2 \pi) $ and $\lim_{x \to 0^+} \arctan_{degree} (x)/x = 180 / \pi $. Adjustment by $2^{-k}/ (2 \pi)$ and $2^{-k}180 / \pi$ requires full-on addition.

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Cycles is pretty straightforward, at least in some cases.

When measuring the rate of rotation of a rotating object, the most common way to do it is to provide some sort of indexing mark, and observe how frequently the mark passes by a reference point. For instance, a flat on a shaft and a switch to detect the flat. Or a bright spot on the shaft, and an optical sensor detects every time the spot passes the sensor.

Or when measuring the frequency of an electrical (sine wave) oscillator, a level detector is used which produces a square wave output, and either the rising or falling edge is detected.

In these cases, one event/count is provided per rotation. Converting that to radians/sec is very often not a useful exercise. A simple cam on a shaft, for instance, will provide one activation of a mechanism per revolution of the shaft. If the shaft speed were measured in rad/sec, it would be necessary to divide by two pi to get the activation frequency of the mechanism. Using cycles is much easier.

An everyday example would be engine rpms vs spark plug firing rate in an automotive engine. If a V4 is operating at 6000 rpm (100 rps), it is trivial to determine the spark plug firing rate: 400 Hz. Adding a conversion factor of two pi to get 628 rps (radians per second rather than revolutions) would not do anybody any favors.

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