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 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 at 1:04

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.

  • $\begingroup$ 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 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. $\endgroup$ – Alexandre Eremenko Apr 1 at 2:04
  • $\begingroup$ Used to have one like that, good times. Now I'm typing deg2rad everywhere. $\endgroup$ – BatWannaBe Apr 1 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. $\endgroup$ – Carl Witthoft Apr 1 at 11:00
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    $\begingroup$ @Carl Witthoft: "radians transcendental"?? They can be rational, or integers:-) $\endgroup$ – Alexandre Eremenko Apr 1 at 11:04

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