# derivatives of $\mathop{sin}\nolimits x$ and $\mathop{cos}\nolimits x$

###### Theorem 1.
 $\frac{d}{dx}(\sin x)=\cos x$
###### Proof.
$\displaystyle\frac{d}{dx}(\sin x)$ $=\displaystyle\lim_{h\to 0}\frac{\sin(x+h)-\sin x}{h}$ $=\displaystyle\lim_{h\to 0}\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}$ by addition formula for sine $=\displaystyle\lim_{h\to 0}\frac{\sin x(\cos h-1)+\cos x\sin h}{h}$ $=\displaystyle\lim_{h\to 0}\left(\sin x\cdot\frac{\cos h-1}{h}+\cos x\cdot% \frac{\sin h}{h}\right)$ $=\displaystyle\sin x\left(\lim_{h\to 0}\frac{\cos h-1}{h}\right)+\cos x\left(% \lim_{h\to 0}\frac{\sin h}{h}\right)$ by this entry (http://planetmath.org/LimitRulesOfFunctions) $=\displaystyle\sin x\cdot 0+\cos x\cdot 1$ by this theorem (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0) and its corollary (http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0) $=\cos x$

###### Theorem 2.
 $\frac{d}{dx}(\cos x)=-\sin x$
###### Proof.
$\displaystyle\frac{d}{dx}(\cos x)$ $=\displaystyle\lim_{h\to 0}\frac{\cos(x+h)-\cos x}{h}$ $=\displaystyle\lim_{h\to 0}\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}$ by addition formula  for cosine (http://planetmath.org/AdditionFormulaForCosine) $=\displaystyle\lim_{h\to 0}\frac{\cos x(\cos h-1)+\sin x\sin h}{h}$ $=\displaystyle\lim_{h\to 0}\left(\cos x\cdot\frac{\cos h-1}{h}-\sin x\cdot% \frac{\sin h}{h}\right)$ $=\displaystyle\cos x\left(\lim_{h\to 0}\frac{\cos h-1}{h}\right)-\sin x\left(% \lim_{h\to 0}\frac{\sin h}{h}\right)$ by this entry (http://planetmath.org/LimitRulesOfFunctions) $=\displaystyle\cos x\cdot 0-\sin x\cdot 1$ by this theorem (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0) and its corollary (http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0) $=-\sin x$

Title derivatives of $\mathop{sin}\nolimits x$ and $\mathop{cos}\nolimits x$ DerivativesOfsinXAndcosX 2013-03-22 16:58:51 2013-03-22 16:58:51 Wkbj79 (1863) Wkbj79 (1863) 8 Wkbj79 (1863) Theorem msc 26A06 msc 26A09 msc 26A03 Derivative2 LimitOfDisplaystyleFracsinXxAsXApproaches0 LimitOfDisplaystyleFrac1CosXxAsXApproaches0 DerivativesOfSineAndCosine