无字证明尝试: 基本三角函数与它们的微分

既然是无字证明, 我就尽量不做任何有意义的必要说明。

Visual trigonometric derivatives

无须多言, 请尽情下载 PDF 版本.

打破惯例, 这次我决定放出代码 (主要使用 TikZ )。 然而相信如此霸气无理的代码风格, 根本没有人能够看得懂。

\def\radiusa{6.0}
\def\radiusb{0.3}
\def\radiusc{5.2}
\def\radiusd{4.5}
\def\desa{1.4}
\def\desb{1.4}
\def\anglea{48}
\def\angleb{138}
\def\anglec{-42}
\def\angleap{55}
\begin{tikzpicture}[x=1.0cm,y=1.0cm,font=\sffamily]
  \clip (-4.0,-4.2) rectangle (24, 10);% 9 * 5
  % \draw[help lines,step=1] (-5,-5) grid (22,12);
  \draw [color=black!20,ultra thin] (-8, 0) -- (8, 0);
  \draw [color=black!20,ultra thin] (0, -8) -- (0, 8);
  \coordinate (O) at (0, 0);
  \coordinate (A) at (\anglea:\radiusa);
  \coordinate (Ap) at (\angleap:\radiusa);
  \coordinate (App) at (\radiusa * cos \angleap, \radiusa * sin \anglea);
  \coordinate (B) at (\radiusa * cos \anglea, 0);
  \coordinate (C) at (0, \radiusa * sin \anglea);
  \coordinate (D) at (\radiusa, 0);
  \coordinate (E) at (0, \radiusa);
  \coordinate (F) at (\radiusa, \radiusa * tan \anglea);
  \coordinate (Fp) at (\radiusa, \radiusa * tan \angleap);
  \coordinate (G) at (\radiusa * cot \anglea, \radiusa);
  \coordinate (Gp) at (\radiusa * cot \angleap, \radiusa);
  \coordinate (H) at (\radiusc * cos \anglec, \radiusc * sin \anglec);
  \coordinate (I) at (\radiusd * cos \angleb, \radiusd * sin \angleb);
  \coordinate (J) at (\radiusa * cot \anglea + \radiusd * cos \angleb, \radiusa + \radiusd * sin \angleb);
  \coordinate (K) at (\radiusa + \radiusc * cos \anglec, \radiusa * tan \anglea + \radiusc * sin \anglec);
  \coordinate (Ba) at (\radiusa * cos \anglea,\radiusb);
  \draw [color=black!20,ultra thick,dashed] (O) circle (\radiusa);
  \draw [color=black,thin] (0:\radiusb) arc(0:\anglea:\radiusb) node[right] {$\theta$};
  \draw [color=black,thick] (O) -- (A) node[midway,below,black] {1};
  \draw [color=red,thick] (A) -- (B) node[midway,right,red] {$\sin$};
  \draw [color=green!80!black,thick] (C) -- (A) node[midway,below,green!80!black] {$\cos$};
  \draw [color=magenta,thick] (D) -- (F) node[midway,right,magenta] {$\tan$};
  \draw [color=cyan!80!black,thick] (E) -- (G) node[midway,above,cyan!80!black] {$\cot$};
  \draw [color=blue!80!black,thick] (O) -- (G);
  \draw [color=orange,thick] (O) -- (F);
  \draw [color=blue!80!black,thick,dashed] (O) -- (I);
  \draw [color=blue!80!black,thick,dashed,name path=lgj] (G) -- (J);
  \draw [color=orange,thick,dashed] (O) -- (H);
  \draw [color=orange,thick,dashed] (F) -- (K);
  \draw [color=white,ultra thin,name path=lfk] (F) -- ++(\angleb:\radiusc);
  \draw [color=blue!80!black,thick,stealth-stealth] (I) -- (J) node[sloped,midway,above,blue!80!black] {$\csc$};
  \draw [color=orange,thick,stealth-stealth] (H) -- (K) node[sloped,midway,below,orange] {$\sec$};
  \draw [color=black,thin] (Ba) -- ++(180:\radiusb) -- ++(270:\radiusb);
  % theta + \delta theta
  \draw [color=black,dashed,name path=lofp] (O) -- (Fp);
  \draw [color=black,ultra thick,fill=red!60] (A)
  -- (Ap) node[midway,above right=-3pt] {$d\theta$}
  -- (App) -- cycle;
  \fill [name intersections={of=lofp and lgj,name=Gpp}];
  \draw [color=black,ultra thick,fill=cyan!50] (G) -- (Gp)
  -- (Gpp-1) -- cycle node[midway,above right=-5pt] {$df$};
  \fill [name intersections={of=lofp and lfk,name=Fpp}];
  \draw [color=black,ultra thick,fill=magenta!50] (F) -- (Fp)
  -- (Fpp-1) -- cycle;
  \node at (5.65, 7.25) {$dg$};
  \draw [color=black!50,thick,dashed] (11.6, -4) -- (11.6, 9);
  % new triangle 1
  \coordinate (NApp) at ($(13.4, -1)$);
  \coordinate (NAp)  at ($(Ap) - (App) + (NApp)$);
  \coordinate (NA)   at ($(A) -(App) + (NApp)$);
  \draw [color=black,ultra thick,fill=red!60] (NA) 
  -- (NAp) node[midway,above right=-2pt] {$d\theta$}
  -- (NApp) node[midway,left] {$d\sin$}
  -- cycle node[midway,below] {$d\cos$};
  \coordinate (NApt) at ($(NAp) + ( \radiusb * sin \anglea, - \radiusb * cos \anglea)$);
  \draw [color=black,thin] (NApt) arc(\anglec:-90:\radiusb);
  \draw [color=black!50,thick,dashed] (12, 1) -- (24, 1);
  % new trangle 2
  \coordinate (NGp)  at ($(13.0, 3)$);
  \coordinate (NG)   at ($(G) - (Gp) + (NGp) $);
  \coordinate (NGpp) at ($(Gpp-1) - (Gp) + (NGp)$);
  \draw [color=black,ultra thick,fill=cyan!50] (NG) 
  -- (NGp)  node[midway,below] {$d\cot$}
  -- (NGpp) node[midway,above left=-2pt] {$d\csc$}
  -- cycle  node[midway,above right=-3pt] {$df$};
  \coordinate (NGpt) at ($(NGp) + (\radiusb, 0)$);
  \draw [color=black,thin] (NGpt) arc(0:\anglea:\radiusb);
  \draw [color=black!50,thick,dashed] (12, 5) -- (24, 5);
  % new trangle 3
  \coordinate (NFpp) at ($(13.1, 7)$);
  \coordinate (NFp)  at ($(Fp) - (Fpp-1) + (NFpp)$);
  \coordinate (NF)   at ($(F) - (Fpp-1) + (NFpp)$);
  \draw [color=black,ultra thick,fill=magenta!50] (NF)
  -- (NFp)  node[midway,right] {$d\tan$}
  -- (NFpp) node[midway,above left] {$d\sec$}
  -- cycle  node[midway,below left=-3pt] {$dg$};
  \coordinate (NFt) at ($(NF) + (0, \radiusb)$);
  \draw [color=black,thin] (NFt) arc(90:\angleb:\radiusb);
  % NOTES
  \fill (6.6, 8) circle (2pt) node [right] {\large $\displaystyle \frac{d\theta}{1} = \frac{df}{\csc \theta} = \frac{dg}{\sec \theta}$};
  \fill (16, -1.5) circle (2pt) node [right] {\large $d\cos\theta = -\sin\theta d\theta$};
  \fill (16, -0.5) circle (2pt) node [right] {\large $d\sin\theta = \cos\theta d\theta$};
  \fill (16, 3.8) circle (2pt) node [right] {\large $\displaystyle d\cot\theta = -\frac{df}{\sin\theta} = -\csc^2\theta d\theta$};
  \fill (16, 2.5) circle (2pt) node [right] {\large $\displaystyle d\csc\theta = -\frac{df}{\tan \theta} = -\csc\theta \cot\theta d\theta$};
  \fill (16, 6.5) circle (2pt) node [right] {\large $\displaystyle d\sec\theta = \tan\theta dg = \sec\theta \tan\theta d\theta$};
  \fill (16, 7.8) circle (2pt) node [right] {\large $\displaystyle d\tan\theta = \frac{dg}{\cos \theta} = \sec^2\theta d\theta$};
\end{tikzpicture}
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