From cc154f9816600dd4ffd87182a218c77110b009b9 Mon Sep 17 00:00:00 2001 From: ulaulaman Date: Mon, 3 Aug 2020 22:59:05 +0200 Subject: [PATCH] Without words: an arctan identity arctan 1/8 + pi/4 + arctan 1/7 + arctan 4/7 = pi /2 --- math/arctan-pi2.tex | 61 ++++++++++++++++++++++++++++++++++++++++ math/pdf/arctan-pi2.pdf | Bin 0 -> 5505 bytes 2 files changed, 61 insertions(+) create mode 100755 math/arctan-pi2.tex create mode 100644 math/pdf/arctan-pi2.pdf diff --git a/math/arctan-pi2.tex b/math/arctan-pi2.tex new file mode 100755 index 0000000..1e7ed0a --- /dev/null +++ b/math/arctan-pi2.tex @@ -0,0 +1,61 @@ +\documentclass{standalone} +% +\usepackage{tikz} +\usetikzlibrary{backgrounds,calc} +\usepackage{tkz-euclide} +\usetkzobj{all} +% +\usepackage{amsmath} +% +\usepackage{xcolor} +% +\definecolor{space}{HTML}{0A2543} +\definecolor{earth}{HTML}{0089FA} +\definecolor{mars}{HTML}{DC7B4E} +\definecolor{dida}{HTML}{FFDE00} +\definecolor{moon}{HTML}{AFAFAF} +% +\usepackage{fontspec} +\setmainfont{Open Dyslexic} +% +\title{Trigonometry: ArcTan pi/2} +\begin{document} + \begin{tikzpicture}[background rectangle/.style={fill=white},show background rectangle] + % + \begin{scope}[scale=8] + \tkzDefPoint(0,0){O} + \tkzDefPoint(56,0){A} + \tkzDefPoint(56,7){B} + \tkzDrawSquare[fill=space](O,A) + % + \tkzDrawPolygon[fill=earth, opacity=0.7](O,A,B) + % + \tkzDefPoint(56,56){C} + \tkzDefPointBy[rotation = center O angle {atan(1/8)}](C) \tkzGetPoint{C1} + \tkzDefPointBy[rotation = center O angle {atan(1/8)}](A) \tkzGetPoint{A1} + \tkzDrawPolygon[fill=mars, opacity=0.7](O,A1,C1) + % + \tkzDefPoint(56,8){D} + \tkzDefPointBy[rotation = center O angle {atan(1/8)+45}](D) \tkzGetPoint{D1} + \tkzDefPointBy[rotation = center O angle {45}](A1) \tkzGetPoint{A2} + \tkzDrawPolygon[fill=dida, opacity=0.7](O,A2,D1) + % + \tkzDefPoint(56,32){E} + \tkzDefPointBy[rotation = center O angle {atan(1/8)+45+atan(1/7)}](E) \tkzGetPoint{E1} + \tkzDefPointBy[rotation = center O angle {atan(1/7)}](A2) \tkzGetPoint{A3} + \tkzDrawPolygon[fill=moon, opacity=0.7](O,A3,E1) + % + \tkzLabelSegment[black,below=5ex,pos=.5](O,A){\fontsize{299}{300}\selectfont 56} + \tkzLabelSegment[black,right=5ex,pos=.5](A,B){\fontsize{299}{300}\selectfont 7} + \tkzLabelSegment[white,above=10ex,pos=.5](O,A1){\fontsize{299}{300}\selectfont 56} + \tkzLabelSegment[white,right=10ex,pos=.5](A1,C1){\fontsize{299}{300}\selectfont 56} + \tkzLabelSegment[white,below=10ex,pos=.5](O,A2){\fontsize{299}{300}\selectfont 56} + \tkzLabelSegment[white,right=10ex,pos=.5](A2,D1){\fontsize{299}{300}\selectfont 8} + \tkzLabelSegment[white,above=20ex,pos=.5](O,A3){\fontsize{299}{300}\selectfont 56} + \tkzLabelSegment[black,right=20ex,pos=.6](A3,E1){\fontsize{299}{300}\selectfont 32} + % + \node at (28,-3) {\textcolor{black}{\fontsize{299}{300}\selectfont Arctan 1/8 + pi/4 + Arctan 1/7 + Arctan 4/7 = pi/2}}; + \end{scope} + % + \end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/math/pdf/arctan-pi2.pdf b/math/pdf/arctan-pi2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5e09347c8f35f70e3549c43497357c07474a1bc7 GIT binary patch literal 5505 zcma)A2|QG77cWs+N}()KHxZL%HnR~L#u8bY8T&dIOffTN#+vflDJrBA(SlS;DN1(A zQc}@oD?&o4H+vd=_o}{p_5I%Oo1Zb~o_o&oocr8!&h!62%GM-f4WuSUM7iZz?bCZN zAKz^d!2$??<*`R(`*r|s%<$vUIpD7!jYlWZJy~9K0KSXP^x^pe1OyJy(*u5dM7TT- zo#rnR(ph3>aij~orgF~ilXya(7y+$ht)A+y_tz`2ne_wKe%HI|wS7Ac=MQWwIj3%3 zuI9v8(vjOF%XyV4POjCx-K+PZ&dhv9{II;xqx+gBQ>Xh)w5>rLozIL;umUo_8zhJENCCS#V%WPkR{`GzlIrzSog9;?VNvC&qJc1OHO zG|o9%ZU1)a+P-n5cjLYAR#mMrwth#*V_%W@^K)OFR(j10X2Rq3gEp;(n|sdkJLZ=q z2_LM?q*a}i@{pWL&GIHn_zehNR-KcVaiL8dEG*`<$=b-l>8;J^VzjNXe2-vL)bX%2 zPb4+2bajRnt_d|tZdIO5I_7vGcAJKl=(?>-Ozx%_*`IhIbRAWYB&Sx3l6E6KSdyCw zbhL`D9XaÐY*bGa+;X+I~eIX2TtaWn%rJ0^Uz$ZCnT6%Fc#9PV&72%la_U+oIOg z@ht;1uq@?UwEyhsX``iXY6IT6KyEw`LqjB@5XHSSL-^Qr{* zhd;ls5`i0DMI$CJ(lw7x4W=w1w(0V@P>01 zc_IIwjg0N%y~B`*(8MhaWJm-c5gG-$;Ys^>!;|*o#;;MJu)dG-O9w^%?eEDjG~ zk>D4D0Dr+R77KcSOa0RsjR8eGQVY~#p>ePXJR~lRgCl?<4l+Ptp_hY4Kn7??i$_5s z7IMRDfua^T4=7?G5f6zjKi?d3@e=;iUVtPl+6sVQdx58WVuWfhyFGDuetZz?6am0t*d@8`0CJ|tKzps z)ZbSE+oIG(sYIpEaRwp6cb7N@IIMqJ{JxD&JUe5SggWX~5JyZRTFWQpTjp13t1chd zUE+T3h8)a5VYf+!Z)R-ny)@_VjoU0ae$LakSaXm61W% zDKs0Iksd}d@K4`eU4KUO{__*I8%oy4>pJ!dR_(khQL(btX=_22V6cx?Q?tRoa_Pao z!EVb?$(Xb`fyV5LcPZU=cM!H^IsH10Ras|FuaxaGQcwGks&K62{$YL&Th5}u@w{;J zO&POQTd$;8p)w0?DxRMkx-~OeTKJ}4?D@9KMnzYw$G-^=1=1ZU7hOMVtsC!r;}QML zJ*dV!xg?!#6TVwxlT+@AcZVK^7k5xEU`9q=)8tlZS6(!8Yq85bIJw^SL8E!MtzmNZ z5u6?Mq=ij*;j=wRDI1-c8AM53#3L%B)Lxm?xuHPYanX{Th+4XN|M-W z&3QtZMEvrSR`o$=nM=aL{G8!}yoB_U*it;W*qOMWX?dFl~rBu}-%^0W1~ zasgJ_PzlB{dbg9;({Xy(A-&K3AvdTww_pVikGHv>ucGc1n=dNnJ5Rp!;HK#1*|Un} z3QJ;czLh+*(ylJqJyH2Cx$g8QV7u^Pd*i#3iFeCM>r*TSBr;7eUT))T>6YiaY|SW@ zm`ez7C@i$azQ0)qYaWhjS4iFZpou#`8FC*o9vVHSE-V>wj*dMWDaG_u+{>CtTrcOl z&vvy=Iq}mP9O7W@)|8YE(y16}#d=G3X`Ye0bkLP$jLaBK$H=%P0jA!|B5z)Fz^hHY z_ODrX4RjTVBbCwMw&BAq<U@#QimC~m^cVPQ7Xj~)(d@KJ%_}Mgg5Sgp@eAj#O$?l>3#LYRwkAbP&z_MGh)5v4 zXi;z~esfz~e&$91_gAq1;)le|2X>!4mC_le zxKn6P{?|_59)-c=?9#EK+vuL0;Ej_N>Da$w_I!wLoDknX=<`!%_sHMF3(x7ZkL{-9IM#<(xS6hMs%6MS|QP4?|s^noYVR}QASdC+BERtzQ;{mIIfEpfb|55e?uYz@ZbkPT*zMBbJ&JiZqJ+j1pu zOGY%1A3hVt_gA*w@_Kq+cT$9$`FYBoLom9eH2DB&daOxrYi!z^qL*vlh7e!%g)3|B zoEO^qVQ~4kPZ!!>!)Sx;&tzhjF1^fFHWrH3(K1kI5sEM!{m@f>KBsOQie3Xt+W~$aY?CK=x+iI7H<+nCaPVO+g@XiD#5_GWqli1gw>lXNt&ZXso z!PpAJ;Qeh}@An(k;|D#F4-|Q_eV9wbBTu|8F4LR;*emDUufJUHk#Y6uF=`o8EtZGP z4<4@qfAN-ONr{wDdwJ%)1x+Y=LyrC?`Ks;dJ<@k!%ll@I6uISEuxTo! zFa@-E|N5y|(;^QkxZBXSJ*Hg}c(|?gBVRnuiFj7rKQ2^C?~FNnqY&jiDaMNJ7tU8Dj>lX)a@M$Nqqgo< z!6O%%+{GGvbL&0%4eeW$<+ra;5FP7?5OzMT5IgoR;Cxuvog&HYRFUp3r3usGjj2)9 z>_|TTF)nL}TH|+1A_o>?&Q(@7@kC?j~^Y36HejL{cQmP4uISL z@IfLVGblt5cF;kb`2!t-xB$X@3&4cD80#&9WdF!owJ^wEU||tbiA{6adwz#h+(-Vp z8`6;zb=q>0nG!KuZPx^4lIx4hgT3D>DLj^yT{lwwI)iXMmP{u89q?*=e-=MPlfUBX z;a)fs78+@}>QrR)oK49qzvD0vU(Km3SskwZVXXOvZDCAyOR$lx>66zo0h=9k*(a%P zhf3M&KO1`VADvw~KchcQpZ{1?%_4ex+SO;e&x>Zo-)Yk;Ilkh_i-tG*kVp}Z5izU$ z^%U2Sl-+E^8-51UhdI>-}c%SbuljOsIyz!RuiWfY2z5t=Wb294xXLf$q%lN zo`ntU49#36yvys!WEijGbx5r%_pw2C^H;NMW4qPG5J@o539uqi5Lhd|fC zs7%U}BQL@}(}HE3fRz3mgk|L42B3CC@9>$fT zO~{;YG`%cJn^w(I$VKNrZOhCjB>3%VY z1*3lqZZQJ>zoUU-SXAR6X;|QMb^t^^Kw<}#Pbe3xww+9G)&dm*>zo0S#85z$2$D5A z41im4yyzTgIaGfx2dJb3ve|xgf2h0>Q2qRR5{T{n=Yv3;k_$joFA(Ym0NkA61uYE3 zKB0B_<%|52kQ&m!*1!Va#rIKu=aR_9r&AyU^a}I@&!<#(1=6T=d+;O+iO@uV;TnQx zQBadp1+_mf87w9VqLr#7Z4?59K;RK*6bhq-#;GAtDhPxMXlKFl`d=ng4vo#Gdx5=) z=EsEyG}Xvn(>T!2&w$2cLZMK78C(GT`yP@yU`yw4!Ks1ou>q~loq&Z&<9}tTZe{L@^>BR^}7yi&41_+ z;5puJb_j5Ne%EQCL9Y6nofZ~kn7`?eXplqxrbA=?OUxLMv;Jm>#Qv#6f~ z{O${?{