# Derivative of the sine function when the argument is measured in degrees

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?

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yes . .a a a aaaoeeo –  sperners lemma Oct 16 '12 at 15:49

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