# Quod Erat Demonstrandum

## 2018/03/05

### 度數弧度微積分

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 12:10 下午
Tags: , ,

（免插聲明：本篇頗無聊，高手見諒）

$\displaystyle \frac{d}{dx}\sin x$　at　$x=0^o$

M2 學生應知

$\displaystyle \frac{d}{dx}\sin x=\cos x$

$\displaystyle \frac{d}{dx}\sin x=\cos 0^o=1$

$\displaystyle \frac{d}{dx}\sin x\ne 1$

0.017432921…

$x$ 以弧度量度，扇形 $ABC$ 的面積是 $\displaystyle \frac{1}{2}\times 1^2\times x=\frac{x}{2}$

$\displaystyle \frac{1}{2}\sin x < \frac{x}{2} < \frac{1}{2}\tan x$

$\displaystyle 1 < \frac{x}{\sin x} < \frac{1}{\cos x}$

$\displaystyle \lim_{x \rightarrow 0}\frac{\sin x}{x}=1$

$\displaystyle \frac{1}{2}\sin x < \frac{x\pi}{360} < \frac{1}{2}\tan x$

$\displaystyle \frac{180}{\pi} < \frac{x}{\sin x} < \frac{180}{\pi} \frac{1}{\cos x}$

$\displaystyle \lim_{x \rightarrow 0}\frac{\sin x}{x}=\frac{\pi}{180}$

$\displaystyle \lim_{x \rightarrow 0}\frac{\sin x^o}{x^o}$

$\displaystyle \lim_{x \rightarrow 0^o}\frac{\sin x}{x}$

$\displaystyle Sin(x)$ 代表輸入的 $x$ 是以度數量度，比如

$\displaystyle Sin(90)=\sin(\frac{\pi}{2}\quad rad)=1$

$\displaystyle \lim_{x \rightarrow 0^o}\frac{\sin x}{x}$

$\displaystyle =\lim_{x \rightarrow 0}\frac{Sin(x)}{x}$

$\displaystyle =\lim_{x \rightarrow 0}\frac{\sin(\frac{\pi}{180}x\quad rad)}{x}$

$\displaystyle =\lim_{x \rightarrow 0}\frac{\sin(\frac{\pi}{180}x\quad rad)}{\frac{\pi}{180}x}\times \frac{\pi}{180}$

$\displaystyle =1\times \frac{\pi}{180}$

$\displaystyle =\frac{\pi}{180}$

$\displaystyle \frac{d}{dx} Sin(x)$

$\displaystyle =\frac{d}{dx} \sin(\frac{\pi}{180}x)$

$\displaystyle =\lim_{h\rightarrow 0}\frac{1}{h}(\sin\frac{\pi}{180}(x+h)-\sin\frac{\pi}{180}x)$

$\displaystyle =\lim_{h\rightarrow 0}\frac{2}{h}\sin(\frac{1}{2}\cdot \frac{\pi}{180}(x+h-x))\cos(\frac{1}{2}\cdot \frac{\pi}{180}(x+h+x))$

$\displaystyle =\lim_{h\rightarrow 0}\frac{2}{h}\sin\frac{\pi}{180}(\frac{h}{2})\cos\frac{\pi}{180}(x+\frac{h}{2})$

$\displaystyle =\lim_{h\rightarrow 0}\frac{\sin\frac{\pi}{180}(\frac{h}{2})}{\frac{h}{2}}\cos\frac{\pi}{180}(x+\frac{h}{2})$

$\displaystyle =\lim_{h\rightarrow 0}\frac{Sin(\frac{h}{2})}{\frac{h}{2}}\lim_{h\rightarrow 0}\cos\frac{\pi}{180}(x+\frac{h}{2})$

$\displaystyle =\frac{\pi}{180}\cdot \cos\frac{\pi}{180}x$

$\displaystyle =\frac{\pi}{180}Cos(x)$

$\displaystyle \frac{d}{dx}\sin x=\frac{\pi}{180}\cos x$

$\displaystyle \frac{d}{dx}\sin x=\frac{\pi}{180}\cos 0^o=\frac{\pi}{180}$

$\displaystyle \frac{d}{dx} Sin(x)=\frac{d}{dx} \sin\frac{\pi}{180} x=\cos\frac{\pi}{180} x\cdot (\frac{\pi}{180} x)'=\frac{\pi}{180}\cos\frac{\pi}{180} x=\frac{\pi}{180}Cos(x)$

$\displaystyle \frac{d^n}{dx^n} Sin(x)=(\frac{\pi}{180})^n Sin(90n+x)$

$\displaystyle Sin(x)=\frac{\pi}{180}x-\frac{(\frac{\pi}{180}x)^3}{3!}+\frac{(\frac{\pi}{180}x)^5}{5!}-\dots$

$\displaystyle \sin1^o=Sin(1)=\frac{\pi}{180}-\frac{(\frac{\pi}{180})^3}{3!}+\frac{(\frac{\pi}{180})^5}{5!}-\dots$

(a) Is $\displaystyle \frac{d}{dx}(x\sin x)|_{x=\frac{\pi}{6}}=\displaystyle \frac{d}{dx}(xSin x)|_{x=30^o}$ ?

(b) Is $\displaystyle \frac{d}{dx}(x^2\tan x)|_{x=\frac{\pi}{4}}=\displaystyle \frac{d}{dx}(x^2Tan x)|_{x=45^o}$ ?

https://johnmayhk.wordpress.com/2010/02/27/circular-argument-on-sinx-over-x/