### Theorem The derivative of the [[Sine Function]] is the [[Cosine Function]]: $ \frac{d}{d x}(\sin x)=\cos x $ #### Proof [[Angle Sum-or-Difference Identities]] [[Table of Common limits]] If $f(x)=\sin x$, then $ \begin{aligned} \left(\sin x\right)^{\prime} &=\lim _{h \rightarrow 0} \frac{\sin (x+h)-\sin x}{h} \\ &=\lim _{h \rightarrow 0} \frac{(\sin x \cos h+\cos x \sin h)-\sin x}{h} \\ &=\lim _{h \rightarrow 0} \frac{\sin x(\cos h-1)+\cos x \sin h}{h} \\ &=\lim _{h \rightarrow 0}\left(\sin x \cdot \frac{\cos h-1}{h}\right)+\lim _{h \rightarrow 0}\left(\cos x \cdot \frac{\sin h}{h}\right) \\ &=\sin x \cdot \lim _{h \rightarrow 0} \frac{\cos h-1}{h}+\cos x \cdot \lim _{h \rightarrow 0} \frac{\sin h}{h}=\sin x \cdot 0+\cos x \cdot 1=\cos x \end{aligned} $