삼각함수의 미분법
Derivatives of Trigonometric Functions
정리
$$ \begin{align*} \sin ^{\prime} x = \cos x \\ \cos ^{\prime} x = - \sin x \\ \tanh ^{\prime} x = \sec^{2} x \end{align*} $$
증명
삼각함수의 덧셈공식 $$ \sin\left( \alpha \pm \beta \right) =\sin\alpha \cos\beta \pm\cos\alpha \sin\beta \\ \cos\left( \alpha \mp \beta \right) =\cos\alpha \cos\beta \pm \sin\alpha \sin\beta $$ 사인함수의 극한
$$ \lim \limits_{x \to 0} \dfrac{\sin x}{x} = 1 $$
코사인함수의 극한 $$ \lim \limits_{x \to 0} \dfrac{1 - \cos x}{x} = 0 $$
$\sin^{\prime} x = \cos x$
정의에 의해,
$$ \begin{align*} & \lim\limits_{h \to 0} \dfrac{\sin (x + h) - \sin x}{h} \\ &= \lim\limits_{h \to 0} \dfrac{\sin x \cos h + \sin h \cos x - \sin x}{h} &\small (\because \text{삼각함수의 덧셈공식}) \\ &= \sin x \lim\limits_{h \to 0} \dfrac{\cos h - 1}{h} + \cos x \lim\limits_{h \to 0} \dfrac{\sin h}{h} \\ &= \sin x \cdot 0 + \cos x \cdot 1 \\ &= \cos x \end{align*} $$
■
$\cos^{\prime} x = -\sin x$
정의에 의해,
$$ \begin{align*} & \lim\limits_{h \to 0} \dfrac{\cos (x + h) - \cos x}{h} \\ &= \lim\limits_{h \to 0} \dfrac{\cos x \cos h - \sin h \sin x - \cos x}{h} &\small (\because \text{삼각함수의 덧셈공식}) \\ &= \cos x \lim\limits_{h \to 0} \dfrac{\cos h - 1}{h} - \sin x \lim\limits_{h \to 0} \dfrac{\sin h}{h} \\ &= \cos x \cdot 0 - \sin x \cdot 1 \\ &= - \sin x \end{align*} $$
■
$\tan^{\prime} x = \sec^{2} x$
$$ \left( \frac{f}{g} \right)^{\prime}(x) = \frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{g^{2}(x)} $$
몫의 미분법에 의해,
$$ \begin{align*} & \tan^{\prime}x \\ &= \left( \dfrac{\sin x}{\cos x} \right)^{\prime} \\ &= \dfrac{\sin^{\prime} x \cos x - \sin x \cos^{\prime} x}{\cos^{2} x} \\ &= \dfrac{\cos x \cos x + \sin x \sin x}{\cos^{2} x} \\ &= \dfrac{1}{\cos^{2} x} \\ &= \left( \dfrac{1}{\cos x} \right)^{2} \\ &= \sec^{2}{x} \end{align*} $$
■