[Math] Derivative of the sine function when the argument is measured in degrees

Question

I'm trying to show that the derivative of $\sin\theta$ is equal to $\pi/180 \cos\theta$ if $\theta$ is measured in degrees. The main idea is that we need to convert $\theta$ to radians to be able to apply the identity $d/dx \sin x = \cos x $. So we need to express
$
\sin \theta$
as
$$
\sin_{deg} \theta = \sin(\pi \theta /180),
$$
where $\sin_{deg}$ is the $\sin$ function that takes degrees as input. Then applying the chain rule yields
$$
d/d\theta [ \sin(\pi\theta/180)] = \cos(\pi \theta/180) \pi/180 = \frac{\pi}{180}\cos_{deg}\theta.
$$
Is this derivation formally correct?

Answer 1

Yes, it is correct, but keep in mind that what you are calculating is:

$$f(x)=\sin(πx/180)$$

$$f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}=\frac{\pi}{180}\cos(\pi x/180),$$ where $x$ is expressed in degrees.