Assuming we do not know radian measure, how do we find the derivative of $\sin(x)$ function when the domain is in degrees?

Question

I stumbled upon this question - Derivative of the sine function when the argument is measured in degrees

where the answer from user Clarinetist uses the fact that $1$ radian = $\pi$/$180$ degrees then apply the Chain Rule to get

$$\dfrac{\text{d}}{\text{d}\theta}\sin_d(\theta) = \dfrac{\pi}{180}\cos_d(\theta)$$

However, I am not satisfied with that answer because it assumes that we already know about radian measure.

Now let's assume we do not know that $1$ radian = $\pi$/$180$ degrees, or in other words, let's assume that the radian measure does not exist. Or in other words, let's pretend that we are in time when Roger Cotes (the inventor of radian) is not born yet. How do we find the derivative of $\sin(x)$ where $x$ is in degrees?

Or is it impossible to determine it without radians?

Answer 1

The main assumption that we use in the derivative of the sine is that $ \sin x \approx x $ for $x \to 0$, if $x$ is in radian.

If $x$ is in degrees, this assumption does not hold. However, if we could conclude that $\sin z \approx (\pi/180)*z $, for small $z$ in degrees, then this is straightforward. How would we go about doing that? Effectively we want to conclude that if $z$ is in degrees, $\lim_{z\to0} \dfrac{\sin z}{z} = \dfrac{\pi}{180}$.

Suppose we use the geometric argument. The arc length follows the rule $$\dfrac{\text{arc length}}{\text{circumference}} = \dfrac{z}{360}$$

which, together with circumference = $2 \pi r$ gives $\text{arc length} = \dfrac{\pi}{180} z $, from there the logic is exactly the same as for radians.

Answer 2

The problem is one of perspective: you're assuming degrees are a natural starting point of known knowledge, when they are not at all, and that "radians" had to actually be invented, when they have really always existed aside from a name. Let me explain.

Imagine a radius of the unit circle is rotating counterclockwise at a constant rate. Suppose you're looking at the shadow the radius casts on the $y$-axis. The essential question is, what is the rate of change of the length of the shadow as the radius points due east?

First, what does the previous question really mean? The most obvious way to make it precise is as follows: if the tip of the radius moves some distance $\Delta_{\mathrm{tip}}$, what is the amount $\Delta_{\mathrm{shadow}}$ that the shadow on the $y$-axis correspondingly moves? Really, if I take a very small amount of movement just as the radius starts pointing due east, what is $\Delta_{\mathrm{shadow}}/\Delta_{\mathrm{tip}}$, roughly? Geometrically it's plain to see that, because the circle is "infinitesimally vertical" at $(1, 0)$, we have $\Delta_{\mathrm{shadow}}/\Delta_{\mathrm{tip}} \approx 1$ here. This is quite literally a geometric proof that $\frac{\mathrm{d}}{\mathrm{d}\theta} \sin\theta|_{\theta=0} = 1$ where $\sin\theta$ is "in radians," aside from a rigorous quantification of the "infinitesimally vertical" statement.

The version of the above setup in degrees is actually the unnatural one. By definition moving one degree is the same as moving $1/360$th of the way around the circumference of the circle. The function "$\sin_d \theta$" is by definition the length of the shadow on the $y$-axis after moving $\theta$-one-three-hundred-sixtieths of the distance around the circle from due east. Then $\frac{\mathrm{d}}{\mathrm{d}\theta} \sin_d\theta$ is really asking about ratio of the amount of movement of the shadow compared to a corresponding change in $\theta$. If we imagine $\theta$ changes by $1$ degree, the tip only moves a distance of $(\text{circumference}/360)$, so the shadow movement is shorter by this factor, i.e. the numerator of our ratio is multiplied by this factor compared to the calculation in the previous paragraph. Hence we see $$ \frac{\mathrm{d}}{\mathrm{d}\theta} \sin_d\theta|_{\theta=0} = \frac{\text{circumference}}{360} \frac{\mathrm{d}}{\mathrm{d}\theta} \sin\theta|_{\theta=0} = \frac{\text{circumference}}{360}. $$ We decided long ago to call the right-hand side "$2\pi/360$" or "$\pi/180$", slightly controversially.