1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
|
\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
input colorbrewer-rgb
ahangle := 30;
beginfig(1);
path h; pair A, B, C, D, O; numeric theta;
h = halfcircle scaled 320;
O = origin;
A = point 4 of h;
B = point 0 of h;
C = point 5/4 of h;
D = (xpart C, ypart A);
2theta = angle C;
draw unitsquare scaled 8 rotated angle (C-D) shifted D withcolor 3/4;
draw unitsquare scaled 8 rotated angle (A-C) shifted C withcolor 3/4;
draw A--C--B withcolor Reds 7 7;
draw O--C--D withcolor Reds 7 7;
drawoptions(withcolor Blues 7 6);
draw h;
label.ulft("$x^2 + y^2 = 1$", point 3 of h);
drawoptions();
primarydef o through p = (1+o/arclength(p))[point 1 of p, point 0 of p] -- (1+o/arclength(p))[point 0 of p, point 1 of p] enddef;
drawarrow 16 through (A--B);
drawarrow 16 through (O--point 2 of h);
dotlabel.bot("$A$", A);
dotlabel.bot("$B$", B);
dotlabel.urt("$C \smash{\;\bigl(\cos2\theta, \sin2\theta\bigr)}$", C);
dotlabel.bot("$D$", D);
dotlabel.llft("$O$", O);
label("$\theta$", 28 dir 1/2 theta shifted A);
label("$2\theta$", 20 dir theta);
label("$x$", B shifted 24 right);
label("$y$", point 2 of h shifted 24 up);
draw thelabel.top("$2\cos\theta$", origin) rotated theta shifted 1/2[A, C];
draw thelabel.top("$2\sin\theta$", origin) rotated (theta-90) shifted 1/2[B, C];
label.bot("$\triangle ACD \sim \triangle ABC$", point 1/2 of bbox currentpicture shifted 24 down);
path bb; bb = bbox currentpicture shifted 12 down;
label.bot(btex \vbox{\openup 4pt\halign{\hfil $#$\hfil\cr
CD \Big/ AC = BC \Big/ AB\cr
\sin 2\theta \big/ 2 \cos\theta = 2 \sin\theta \big/ 2\cr
\sin 2\theta = 2\sin\theta \cos\theta\cr}} etex, point 1/4 of bb);
label.bot(btex \vbox{\openup 4pt\halign{\hfil $#$\hfil\cr
AD \Big/ AC = AC \Big/ AB\cr
\bigl(1 + \cos 2\theta \bigr) \big/ 2 \cos\theta = 2 \cos\theta \big/ 2\cr
\cos 2\theta = 2\cos^2\theta - 1\cr}} etex, point 3/4 of bb);
endfig;
\end{mplibcode}
\end{document}
|