PlanetMath: derivatives of $\sin x$ and $\cos x$

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derivatives of $\sin x$ and $\cos x$

(Theorem)

Theorem $$\frac{d}{dx} (\sin x)=\cos x$$

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
	$=\displaystyle \sin x \cdot 0+\cos x \cdot 1$ by this theorem and its corollary
	$=\cos x$

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Theorem $$\frac{d}{dx} (\cos x)=-\sin x$$

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
	$=\displaystyle \lim_{h \to 0} \frac{\cos x (\cos h-1)+\sin x \sin h}{h}$
	$=\displaystyle \lim_{h \to 0} \left( \cos x \cdot \frac{\cos h-1}{h} -\sin x \cdot \frac{\sin h}{h} \right)$
	$=\displaystyle \cos x \left( \lim_{h \to 0} \frac{\cos h-1}{h} \right) -\sin x \left( \lim_{h \to 0} \frac{\sin h}{h} \right)$ by this entry
	$=\displaystyle \cos x \cdot 0-\sin x \cdot 1$ by this theorem and its corollary
	$=-\sin x$

$\qedsymbol$

"derivatives of $\sin x$ and $\cos x$ " is owned by Wkbj79.

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See Also: derivative, limit of $\displaystyle \frac{\sin x}{x}$ as $x$

approaches 0, limit of $\displaystyle \frac{1-\cos x}{x}$ as $x$

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Cross-references: addition formula for sine
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This is version 5 of derivatives of $\sin x$ and $\cos x$ , born on 2007-04-24, modified 2007-04-25.
Object id is 9257, canonical name is DerivativesOfSinXAndCosX.
Accessed 1382 times total.

Classification:

AMS MSC:	26A03 (Real functions :: Functions of one variable :: Foundations: limits and generalizations, elementary topology of the line)
	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	26A09 (Real functions :: Functions of one variable :: Elementary functions)

Pending Errata and Addenda

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