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三角函數的微分

總之一個因緣際會複習微積分,所以寫下這篇文章。
本篇文章部分要仰賴一些先備知識:

$\displaystyle{ \lim_{\theta \to 0} \frac{sin(\theta)}{\theta} = 1 }$

(證明可以參考:sinx/xについて覚えておくべき2つのこと

題外話,在 tutor 板上出現過為什麼不可以用羅必達法則(l’Hôpital’s Rule)證明的問題。
有興趣可以自己前去觀看:[討論] limsinx/x=1 when x→0

另一個部分是:

$\displaystyle{ \lim_{\theta \to 0} \frac{cos(\theta) - 1}{\theta} = 0 }$

推倒如下。

$\displaystyle{ \lim_{\theta \to 0} \frac{cos(\theta) - 1}{\theta} = \lim_{\theta \to 0} \frac{cos(\theta) - 1}{\theta} \cdot \frac{cos(\theta) + 1}{cos(\theta) + 1}}$

$\displaystyle{ = \lim_{\theta \to 0} \frac{cos^{2}(\theta) - 1^{2}}{\theta [cos(\theta) + 1]} = \lim_{\theta \to 0} \frac{-sin^{2}(\theta)}{\theta [cos(\theta) + 1]}}$

$\displaystyle{ = - \lim_{\theta \to 0} \frac{sin(\theta)}{\theta}\frac{sin(\theta)}{cos(\theta) + 1} = - \{ \lim_{\theta \to 0} \frac{sin(\theta)}{\theta} \cdot \lim_{\theta \to 0} \frac{sin(\theta)}{cos(\theta) + 1} \} }$

$\displaystyle{ = - \{ 1 \cdot \frac{0}{1 + 1} \} = 0 }$ 

sin(x) 正弦

利用微分定義推導:

$\displaystyle{ \frac{d}{dx} sin(x) = \lim_{h \to 0} \frac{sin(x + h) - sin(x)}{h} }$

$\displaystyle{ = \lim_{h \to 0} \frac{sin(x)cos(h) + cos(x)sin(h) - sin(x)}{h} \text{(※ 正弦和角公式)} }$

$\displaystyle{ = \lim_{h \to 0} \frac{sin(x)cos(h) - sin(x) + cos(x)sin(h)}{h} }$

$\displaystyle{ = \lim_{h \to 0} \frac{sin(x) [cos(h) - 1]}{h} + \lim_{h \to 0} \frac{cos(x)sin(h)}{h} }$

$\displaystyle{ = sin(x) \cdot \lim_{h \to 0} \frac{cos(h) - 1}{h} + cos(x) \cdot \lim_{h \to 0} \frac{sin(h)}{h} }$

$\displaystyle{ = sin(x) \cdot 0 + cos(x) \cdot 1 = cos(x) }$

cos(x) 餘弦

利用微分定義推導:

$\displaystyle{ \frac{d}{dx} cos(x) = \lim_{h \to 0} \frac{cos(x + h) - cos(x)}{h} }$

$\displaystyle{ = \lim_{h \to 0} \frac{cos(x)cos(h) - sin(x)sin(h) - cos(x)}{h} \text{(※ 餘弦和角公式)} }$

$\displaystyle{ = \lim_{h \to 0} \frac{cos(x)cos(h) - cos(x) - sin(x)sin(h)}{h} }$

$\displaystyle{ = \lim_{h \to 0} \frac{cos(x) [cos(h) - 1]}{h} - \lim_{h \to 0} \frac{sin(x)sin(h)}{h} }$

$\displaystyle{ = cos(x) \cdot \lim_{h \to 0} \frac{cos(h) - 1}{h} - sin(x) \cdot \lim_{h \to 0} \frac{sin(h)}{h} }$

$\displaystyle{ = cos(x) \cdot 0 - sin(x) \cdot 1 = - sin(x) }$

tan(x) 正切

由 $tan(x) = \frac{sin(x)}{cos(x)}$ 開始:

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

$ = \frac{\frac{d}{dx}sin(x) \cdot cos(x) - sin(x) \cdot \frac{d}{dx}cos(x)}{cos^{2}(x)} \text{(※ 微分除法公式)}$

$ = \frac{cos(x) \cdot cos(x) - sin(x) \cdot [-sin(x)]}{cos^{2}(x)} = \frac{cos^{2}(x) + sin^{2}(x)}{cos^{2}(x)}$

$ = \frac{1}{cos^{2}(x)} = sec^{2}(x)$

(※ 使用了 $sin^{2}(x) + cos^{2}(x) = 1$ 這條式子,證明請參考:sin^2+cos^2=1

csc(x) 餘割

由 $csc(x) = \frac{1}{sin(x)}$ 開始:

$\frac{d}{dx} csc(x) = \frac{d}{dx} \frac{1}{sin(x)}$

$ = \frac{\frac{d}{dx} 1 \cdot sin(x) - 1 \cdot \frac{d}{dx} sin(x)}{sin^{2}(x)} \text{(※ 微分除法公式)}$

$ = \frac{0 \cdot sin(x) - 1 \cdot cos(x)}{sin^{2}(x)} = - \frac{cos(x)}{sin^{2}(x)}$

$ = - \frac{cos(x)}{sin(x)} \cdot \frac{1}{sin(x)} = - cot(x) csc(x)$

sec(x) 正割

由 $sec(x) = \frac{1}{cos(x)}$ 開始:

$\frac{d}{dx} sec(x) = \frac{d}{dx} \frac{1}{cos(x)}$

$ = \frac{\frac{d}{dx} 1 \cdot cos(x) - 1 \cdot \frac{d}{dx} cos(x)}{cos^{2}(x)} \text{(※ 微分除法公式)}$

$ = \frac{0 \cdot cos(x) - 1 \cdot [-sin(x)] }{cos^{2}(x)} = \frac{sin(x)}{cos^{2}(x)}$

$ = \frac{sin(x)}{cos(x)} \cdot \frac{1}{cos(x)} = tan(x) sec(x)$

cot(x) 餘切

由 $cot(x) = \frac{cos(x)}{sin(x)}$ 開始:

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

$ = \frac{\frac{d}{dx}cos(x) \cdot sin(x) - cos(x) \cdot \frac{d}{dx} sin(x)}{sin^{2}(x)} \text{(※ 微分除法公式)}$

$ = \frac{[-sin(x)] \cdot sin(x) - cos(x) \cdot cos(x)}{sin^{2}(x)} = \frac{ - sin^{2}(x) - cos^{2}(x)}{sin^{2}(x)}$

$ = - \{\frac{sin^{2}(x) + cos^{2}(x)}{sin^{2}(x)}\} = - \frac{1}{sin^{2}(x)} = -csc^{2}(x)$

(※ 跟 $tan(x)$ 一樣使用了 $sin^{2}(x) + cos^{2}(x) = 1$ 這條式子,證明請參考上面的資料。)

其他

可以發現一些手法,除了 $sin(x)$ 以及 $cos(x)$ 要從定義來之外,
其他的有兩兩對應的步驟,像是 $tan(x)$ 跟 $cot(x)$ 就很類似、而 $csc(x)$ 跟 $sec(x)$ 也很相像。