Chapter 2.5, Problem 49ES

The derivative formulas for $\sin x,\cos x,\tan x,\cot x,\sec x$ , and $\csc x$ were obtained under the assumption that $x$ is measured in radians. If $x$ is measured in degrees, then

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=\frac{\pi }{180}$

(See Exercise $39$ of Section $1.6$ ). Use this result to prove that if $x$ is measured in degrees, then

(a) $\frac{d}{dx}\left[\sin x\right]=\frac{\pi }{180}\cos x$

(b) $\frac{d}{dx}\left[\cos x\right]=-\frac{\pi }{180}\sin x$