大阪大学 後期理系 1989年度 問5

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解答作成者: 森 宏征

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入試情報

大学名 大阪大学
学科・方式 後期理系
年度 1989年度
問No 問5
学部 理学部 ・ 医学部 ・ 歯学部 ・ 薬学部 ・ 工学部 ・ 基礎工学部
カテゴリ 三角関数
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{vector3} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{color} \ExecuteOptions{usename} \usepackage{graphicx} \usepackage{custom_mori} \usepackage{pifont} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  $D$ は$xy$平面上の, 不等式 $x^2 + y^2 \leqq 1,\,\,\,x \geqq 0,\,\,\,y \geqq 0$ で表される領域とする.$f(\theta)\,\,\,(0 \leqq \theta < 2\pi)$ を, $x,\,\,y$ の関数 $x\cos\theta + y\sin\theta$ が $D$ にとる最大値とする. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  解答用紙の表にしたがって $f(\theta)$ を求め, それぞれの欄(省略)に記入せよ. \item  $f(\theta)$ のグラフの概形を, 解答用紙の図(省略)中にかきこめ. \end{enumerate} \end{FRAME} \noindent{\color[named]{BurntOrange}\bfseries \fbox{解答}} \vspace{-2mm} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $x^2 + y^2 \leqq 1,\,\,\,x \geqq 0,\,\,\,y \geqq 0$ より \begin{gather*} x = r\cos\pphi,\quad y = r\sin\pphi \tag*{$\cdott\MARU{1}$} \\ 0 \leqq r \leqq 1,\quad 0 \leq \pphi \leqq \frac{\pi}{2} \tag*{$\cdott\MARU{2}$} \end{gather*} とおける. \MARU{1}より \begin{align*} x\cos\theta + y\sin\theta &= r\cos\pphi \cos\theta + r\sin\pphi \sin\theta \\ &= r\cos(\pphi - \theta) = g(\pphi) \quad \mbox{\footnotesize (とする)} \end{align*} まず,$r$ を固定し $\pphi$ のみを変化させたときの最大値を求め, 次に $r$ を変化させて真の最大値$(= f(\theta))$を求める. \begin{enumerate} \item[(i)] $-\dfrac{\pi}{2} \leqq -\theta \leqq 0\,\,\, \left(0 \leqq \theta \leqq \dfrac{\pi}{2}\right)$ \smallskip のとき  \MARU{2}より \begin{align*} -\theta \leqq \pphi - \theta \leqq \frac{\pi}{2} - \theta \tag*{$\cdott\MARU{2}'$} \end{align*} $r$ を固定すれば,$\pphi = \theta$ で $g(\pphi)$ は 最大値 \[ g(\theta) = r \] をとる. $r$ を変化させて真の最大値を求めると, \begin{align*} f(\theta) = (rの最大値) = 1 \tag*{$\cdott\MARU{3}$} \end{align*} \item[(ii)] $-\pi \leqq -\theta \leqq -\dfrac{\pi}{2}\,\,\, \left(\dfrac{\pi}{2} \leqq \theta \leqq \pi \right)$ のとき \smallskip  $\MARU{2}'$より $r$ を固定すれば, $\pphi - \theta = \dfrac{\pi}{2} - \theta$ つまり % $\pphi = \dfrac{\pi}{2}$ で $g(\pphi)$ は最大値 \begin{align*} g\!\left(\frac{\pi}{2} \right) = r\cos\left(\frac{\pi}{2} - \theta \right) = r\sin\theta \end{align*} をとる. $\sin\theta \geqq 0$ に注意する. $r$ を変化させて真の最大値を求めると, \begin{align*} f(\theta) = 1 \cdot \sin\theta = \sin\theta \tag*{$\cdott\MARU{4}$} \end{align*} \newpage \item[(iii)] $-\dfrac{3}{2}\pi \leqq -\theta \leqq -\pi\,\,\, \left(\pi \leqq \theta \leqq \dfrac{3}{2}\pi \right)$ のとき \smallskip  $\MARU{2}'$より $r$ を固定すれば,$g(\pphi)$ は \begin{align*} \pphi - \theta = -\theta\,\,\,または\,\,\,\frac{\pi}{2} - \theta \qquad \therefore \,\,\, \pphi = 0\,\,\,または\,\,\,\frac{\pi}{2} \end{align*} のとき最大となり,最大値 \begin{gather*} g(0) = r\cos(-\theta) = r\cos\theta \,\,(\leqq 0) \\ または\,\,\, g\!\left(\frac{\pi}{2} \right) = r\sin\theta \,\,(\leqq 0) \end{gather*} をとる. $r$ を変化させて真の最大値を求める. いずれの場合も $r = 0$ で最大になり, \begin{align*} f(\theta) = 0 \tag*{$\cdott\MARU{5}$} \end{align*} \item[(iv)] $-2\pi \leqq -\theta \leqq -\dfrac{3}{2}\pi\,\,\, \left(\dfrac{3}{2}\pi \leqq \theta \leqq 2\pi \right)$ のとき \smallskip  $\MARU{2}'$より $r$ を固定すれば, $\pphi - \theta = -\theta$ つまり $\pphi = 0$ で $g(\pphi)$ は最大値 \begin{align*} g(0) = r\cos(-\theta) = r\cos\theta \end{align*} をとる.$\cos\theta \geqq 0$ に留意する. $r$ を変化させて真の最大値を求めると, \begin{align*} f(\theta) = 1 \cdot \cos\theta = \cos\theta \tag*{$\cdott\MARU{6}$} \end{align*} \end{enumerate} \begin{center} %\input{osaka89t2s_zu_1} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 37.3800, 36.2900)( 8.7000,-40.6800) % CIRCLE 2 0 3 0 % 4 1637 1333 1637 1753 1637 1753 1637 1753 % \special{pn 8}% \special{ar 1638 1334 420 420 0.0000000 6.2831853}% % VECTOR 2 0 3 0 % 2 1007 1333 2267 1333 % \special{pn 8}% \special{pa 1008 1334}% \special{pa 2268 1334}% \special{fp}% \special{sh 1}% \special{pa 2268 1334}% \special{pa 2200 1314}% \special{pa 2214 1334}% \special{pa 2200 1354}% \special{pa 2268 1334}% \special{fp}% % VECTOR 2 0 3 0 % 2 1637 1963 1637 703 % \special{pn 8}% \special{pa 1638 1964}% \special{pa 1638 704}% \special{fp}% \special{sh 1}% \special{pa 1638 704}% \special{pa 1618 770}% \special{pa 1638 756}% \special{pa 1658 770}% \special{pa 1638 704}% \special{fp}% % VECTOR 1 0 3 0 % 2 1637 1333 1847 1701 % \special{pn 13}% \special{pa 1638 1334}% \special{pa 1848 1702}% \special{fp}% \special{sh 1}% \special{pa 1848 1702}% \special{pa 1832 1634}% \special{pa 1822 1656}% \special{pa 1798 1654}% \special{pa 1848 1702}% \special{fp}% % VECTOR 1 0 3 0 % 2 1637 1333 2004 1123 % \special{pn 13}% \special{pa 1638 1334}% \special{pa 2004 1124}% \special{fp}% \special{sh 1}% \special{pa 2004 1124}% \special{pa 1936 1140}% \special{pa 1958 1150}% \special{pa 1956 1174}% \special{pa 2004 1124}% \special{fp}% % POLYGON 2 0 3 0 % 5 1639 1334 1704 1295 1743 1359 1677 1399 1639 1334 % \special{pn 8}% \special{pa 1640 1334}% \special{pa 1704 1296}% \special{pa 1744 1360}% \special{pa 1678 1400}% \special{pa 1640 1334}% \special{fp}% % STR 2 0 3 0 % 3 1500 695 1500 800 2 0 % $y$ \put(15.0000,-8.0000){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 2298 1260 2298 1365 2 0 % $x$ \put(22.9800,-13.6500){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 2078 1370 2078 1475 2 0 % {\small 1} \put(20.7800,-14.7500){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 1668 777 1668 882 2 0 % {\small 1} \put(16.6800,-8.8200){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 1430 1785 1430 1890 2 0 % {\small $-1$} \put(14.3000,-18.9000){\makebox(0,0)[lb]{{\small $-1$}}}% % STR 2 0 3 0 % 3 1010 1375 1010 1480 2 0 % {\small $-1$} \put(10.1000,-14.8000){\makebox(0,0)[lb]{{\small $-1$}}}% % CIRCLE 2 0 3 0 % 4 1637 3438 1637 3858 1637 3858 1637 3858 % \special{pn 8}% \special{ar 1638 3438 420 420 0.0000000 6.2831853}% % VECTOR 2 0 3 0 % 2 1007 3438 2267 3438 % \special{pn 8}% \special{pa 1008 3438}% \special{pa 2268 3438}% \special{fp}% \special{sh 1}% \special{pa 2268 3438}% \special{pa 2200 3418}% \special{pa 2214 3438}% \special{pa 2200 3458}% \special{pa 2268 3438}% \special{fp}% % VECTOR 2 0 3 0 % 2 1637 4068 1637 2808 % \special{pn 8}% \special{pa 1638 4068}% \special{pa 1638 2808}% \special{fp}% \special{sh 1}% \special{pa 1638 2808}% \special{pa 1618 2876}% \special{pa 1638 2862}% \special{pa 1658 2876}% \special{pa 1638 2808}% \special{fp}% % VECTOR 1 0 3 0 % 2 3943 3440 4314 3234 % \special{pn 13}% \special{pa 3944 3440}% \special{pa 4314 3234}% \special{fp}% \special{sh 1}% \special{pa 4314 3234}% \special{pa 4246 3250}% \special{pa 4268 3260}% \special{pa 4266 3284}% \special{pa 4314 3234}% \special{fp}% % VECTOR 1 0 3 0 % 2 3943 3440 3737 3071 % \special{pn 13}% \special{pa 3944 3440}% \special{pa 3738 3072}% \special{fp}% \special{sh 1}% \special{pa 3738 3072}% \special{pa 3752 3140}% \special{pa 3764 3118}% \special{pa 3788 3120}% \special{pa 3738 3072}% \special{fp}% % POLYGON 2 0 3 0 % 5 3943 3440 3906 3375 3971 3337 4010 3402 3943 3440 % \special{pn 8}% \special{pa 3944 3440}% \special{pa 3906 3376}% \special{pa 3972 3338}% \special{pa 4010 3402}% \special{pa 3944 3440}% \special{fp}% % STR 2 0 3 0 % 3 1595 2677 1595 2782 2 0 % $y$ \put(15.9500,-27.8200){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 2298 3370 2298 3475 2 0 % $x$ \put(22.9800,-34.7500){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 2078 3470 2078 3575 2 0 % {\small 1} \put(20.7800,-35.7500){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 1668 2882 1668 2987 2 0 % {\small 1} \put(16.6800,-29.8700){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 1658 3869 1658 3974 2 0 % {\small $-1$} \put(16.5800,-39.7400){\makebox(0,0)[lb]{{\small $-1$}}}% % STR 2 0 3 0 % 3 1007 3307 1007 3412 2 0 % {\small $-1$} \put(10.0700,-34.1200){\makebox(0,0)[lb]{{\small $-1$}}}% % VECTOR 1 0 3 0 % 2 1637 3449 1480 3054 % \special{pn 13}% \special{pa 1638 3450}% \special{pa 1480 3054}% \special{fp}% \special{sh 1}% \special{pa 1480 3054}% \special{pa 1486 3124}% \special{pa 1500 3104}% \special{pa 1524 3110}% \special{pa 1480 3054}% \special{fp}% % VECTOR 1 0 3 0 % 2 1637 3449 1245 3605 % \special{pn 13}% \special{pa 1638 3450}% \special{pa 1246 3606}% \special{fp}% \special{sh 1}% \special{pa 1246 3606}% \special{pa 1314 3600}% \special{pa 1296 3586}% \special{pa 1300 3562}% \special{pa 1246 3606}% \special{fp}% % POLYGON 2 0 3 0 % 5 1637 3449 1568 3478 1539 3409 1607 3378 1637 3449 % \special{pn 8}% \special{pa 1638 3450}% \special{pa 1568 3478}% \special{pa 1540 3410}% \special{pa 1608 3378}% \special{pa 1638 3450}% \special{fp}% % CIRCLE 2 0 3 0 % 4 3947 1338 3947 1758 3947 1758 3947 1758 % \special{pn 8}% \special{ar 3948 1338 420 420 0.0000000 6.2831853}% % VECTOR 2 0 3 0 % 2 3317 1338 4577 1338 % \special{pn 8}% \special{pa 3318 1338}% \special{pa 4578 1338}% \special{fp}% \special{sh 1}% \special{pa 4578 1338}% \special{pa 4510 1318}% \special{pa 4524 1338}% \special{pa 4510 1358}% \special{pa 4578 1338}% \special{fp}% % VECTOR 2 0 3 0 % 2 3947 1968 3947 708 % \special{pn 8}% \special{pa 3948 1968}% \special{pa 3948 708}% \special{fp}% \special{sh 1}% \special{pa 3948 708}% \special{pa 3928 776}% \special{pa 3948 762}% \special{pa 3968 776}% \special{pa 3948 708}% \special{fp}% % STR 2 0 3 0 % 3 3820 705 3820 810 2 0 % $y$ \put(38.2000,-8.1000){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 4608 1270 4608 1375 2 0 % $x$ \put(46.0800,-13.7500){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 4388 1370 4388 1475 2 0 % {\small 1} \put(43.8800,-14.7500){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 3978 782 3978 887 2 0 % {\small 1} \put(39.7800,-8.8700){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 3740 1795 3740 1900 2 0 % {\small $-1$} \put(37.4000,-19.0000){\makebox(0,0)[lb]{{\small $-1$}}}% % STR 2 0 3 0 % 3 3317 1207 3317 1312 2 0 % {\small $-1$} \put(33.1700,-13.1200){\makebox(0,0)[lb]{{\small $-1$}}}% % VECTOR 1 0 3 0 % 2 3947 1338 3569 1530 % \special{pn 13}% \special{pa 3948 1338}% \special{pa 3570 1530}% \special{fp}% \special{sh 1}% \special{pa 3570 1530}% \special{pa 3638 1518}% \special{pa 3618 1506}% \special{pa 3620 1482}% \special{pa 3570 1530}% \special{fp}% % VECTOR 1 0 3 0 % 2 3947 1338 4139 1715 % \special{pn 13}% \special{pa 3948 1338}% \special{pa 4140 1716}% \special{fp}% \special{sh 1}% \special{pa 4140 1716}% \special{pa 4128 1648}% \special{pa 4116 1668}% \special{pa 4092 1666}% \special{pa 4140 1716}% \special{fp}% % POLYGON 2 0 3 0 % 5 3947 1338 3982 1404 3916 1440 3879 1374 3947 1338 % \special{pn 8}% \special{pa 3948 1338}% \special{pa 3982 1404}% \special{pa 3916 1440}% \special{pa 3880 1374}% \special{pa 3948 1338}% \special{fp}% % CIRCLE 2 0 3 0 % 4 3947 3438 3947 3858 3947 3858 3947 3858 % \special{pn 8}% \special{ar 3948 3438 420 420 0.0000000 6.2831853}% % VECTOR 2 0 3 0 % 2 3317 3438 4577 3438 % \special{pn 8}% \special{pa 3318 3438}% \special{pa 4578 3438}% \special{fp}% \special{sh 1}% \special{pa 4578 3438}% \special{pa 4510 3418}% \special{pa 4524 3438}% \special{pa 4510 3458}% \special{pa 4578 3438}% \special{fp}% % VECTOR 2 0 3 0 % 2 3947 4068 3947 2808 % \special{pn 8}% \special{pa 3948 4068}% \special{pa 3948 2808}% \special{fp}% \special{sh 1}% \special{pa 3948 2808}% \special{pa 3928 2876}% \special{pa 3948 2862}% \special{pa 3968 2876}% \special{pa 3948 2808}% \special{fp}% % STR 2 0 3 0 % 3 4608 3370 4608 3475 2 0 % $x$ \put(46.0800,-34.7500){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 4388 3470 4388 3575 2 0 % {\small 1} \put(43.8800,-35.7500){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 3978 2882 3978 2987 2 0 % {\small 1} \put(39.7800,-29.8700){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 3968 3869 3968 3974 2 0 % {\small $-1$} \put(39.6800,-39.7400){\makebox(0,0)[lb]{{\small $-1$}}}% % STR 2 0 3 0 % 3 3330 3465 3330 3570 2 0 % {\small $-1$} \put(33.3000,-35.7000){\makebox(0,0)[lb]{{\small $-1$}}}% % STR 2 0 3 0 % 3 3348 1527 3348 1632 2 0 % {\small$-\theta$} \put(33.4800,-16.3200){\makebox(0,0)[lb]{{\small$-\theta$}}}% % STR 2 0 3 0 % 3 1820 1715 1820 1820 2 0 % {\small$-\theta$} \put(18.2000,-18.2000){\makebox(0,0)[lb]{{\small$-\theta$}}}% % STR 2 0 3 0 % 3 4090 1825 4090 1930 2 0 % {\small$\frac{\pi}{2}-\theta$} \put(40.9000,-19.3000){\makebox(0,0)[lb]{{\small$\frac{\pi}{2}-\theta$}}}% % STR 2 0 3 0 % 3 2025 1034 2025 1139 2 0 % {\small$\frac{\pi}{2}-\theta$} \put(20.2500,-11.3900){\makebox(0,0)[lb]{{\small$\frac{\pi}{2}-\theta$}}}% % STR 2 0 3 0 % 3 1301 2919 1301 3024 2 0 % {\small$-\theta$} \put(13.0100,-30.2400){\makebox(0,0)[lb]{{\small$-\theta$}}}% % STR 2 0 3 0 % 3 870 3706 870 3811 2 0 % {\small$\frac{\pi}{2}-\theta$} \put(8.7000,-38.1100){\makebox(0,0)[lb]{{\small$\frac{\pi}{2}-\theta$}}}% % STR 2 0 3 0 % 3 3453 2950 3453 3055 2 0 % {\small$\frac{\pi}{2}-\theta$} \put(34.5300,-30.5500){\makebox(0,0)[lb]{{\small$\frac{\pi}{2}-\theta$}}}% % STR 2 0 3 0 % 3 4325 3139 4325 3244 2 0 % {\small$-\theta$} \put(43.2500,-32.4400){\makebox(0,0)[lb]{{\small$-\theta$}}}% % STR 2 0 3 0 % 3 3905 2677 3905 2782 2 0 % $y$ \put(39.0500,-27.8200){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 1007 504 1007 609 2 0 % {\footnotesize (i)の場合} \put(10.0700,-6.0900){\makebox(0,0)[lb]{{\footnotesize (i)の場合}}}% % STR 2 0 3 0 % 3 3317 504 3317 609 2 0 % {\footnotesize (ii)の場合} \put(33.1700,-6.0900){\makebox(0,0)[lb]{{\footnotesize (ii)の場合}}}% % STR 2 0 3 0 % 3 3317 2499 3317 2604 2 0 % {\footnotesize (iv)の場合} \put(33.1700,-26.0400){\makebox(0,0)[lb]{{\footnotesize (iv)の場合}}}% % STR 2 0 3 0 % 3 1007 2499 1007 2604 2 0 % {\footnotesize (iii)の場合} \put(10.0700,-26.0400){\makebox(0,0)[lb]{{\footnotesize (iii)の場合}}}% \end{picture}% \end{center}  (i)\,~\,(iv)より \begin{align*} \hspace{-2.5zw} f(\theta) = \textcolor{red}{\boldsymbol{ \left\{ \begin{array}{ll} 1 & \quad \left(0 \leqq \theta \leqq \dfrac{\pi}{2} \right) \\[3.5mm] \sin\theta & \quad \left(\dfrac{\pi}{2} \leqq \theta \leqq \pi \right) \\[3.5mm] 0 & \quad \left(\pi \leqq \theta \leqq \dfrac{3\pi}{2} \right) \\[4mm] \cos\theta & \quad \left(\dfrac{3\pi}{2} \leqq \theta \leqq 2\pi \right) \end{array} \right. }} \tag*{$\Ans$} \end{align*} \item  (1)の結果より $y = f(\theta)$ のグラフは下図の太線部分のようになる. \begin{center} %\input{osaka89t2s_zu_2} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 25.0000, 8.6000)( 9.0000,-21.8000) % STR 2 0 3 0 % 3 1170 1309 1170 1320 4 1600 % $y$ \put(11.7000,-13.2000){\makebox(0,0)[rt]{$y$}}% % VECTOR 2 0 3 0 % 2 1200 2180 1202 1320 % \special{pn 8}% \special{pa 1200 2180}% \special{pa 1202 1320}% \special{fp}% \special{sh 1}% \special{pa 1202 1320}% \special{pa 1182 1388}% \special{pa 1202 1374}% \special{pa 1222 1388}% \special{pa 1202 1320}% \special{fp}% % VECTOR 2 0 3 0 % 2 958 2004 3358 2004 % \special{pn 8}% \special{pa 958 2004}% \special{pa 3358 2004}% \special{fp}% \special{sh 1}% \special{pa 3358 2004}% \special{pa 3292 1984}% \special{pa 3306 2004}% \special{pa 3292 2024}% \special{pa 3358 2004}% \special{fp}% % FUNC 1 0 3 0 % 9 958 1370 3358 2130 1198 2004 2158 2004 1198 1560 1198 1370 1678 2130 0 5 1 0 % 1 \special{pn 13}% \special{pa 956 1560}% \special{pa 960 1560}% \special{pa 966 1560}% \special{pa 970 1560}% \special{pa 976 1560}% \special{pa 980 1560}% \special{pa 986 1560}% \special{pa 990 1560}% \special{pa 996 1560}% \special{pa 1000 1560}% \special{pa 1006 1560}% \special{pa 1010 1560}% \special{pa 1016 1560}% \special{pa 1020 1560}% \special{pa 1026 1560}% \special{pa 1030 1560}% \special{pa 1036 1560}% \special{pa 1040 1560}% \special{pa 1046 1560}% \special{pa 1050 1560}% \special{pa 1056 1560}% \special{pa 1060 1560}% \special{pa 1066 1560}% \special{pa 1070 1560}% \special{pa 1076 1560}% \special{pa 1080 1560}% \special{pa 1086 1560}% \special{pa 1090 1560}% \special{pa 1096 1560}% \special{pa 1100 1560}% \special{pa 1106 1560}% \special{pa 1110 1560}% \special{pa 1116 1560}% \special{pa 1120 1560}% \special{pa 1126 1560}% \special{pa 1130 1560}% \special{pa 1136 1560}% \special{pa 1140 1560}% \special{pa 1146 1560}% \special{pa 1150 1560}% \special{pa 1156 1560}% \special{pa 1160 1560}% \special{pa 1166 1560}% \special{pa 1170 1560}% \special{pa 1176 1560}% \special{pa 1180 1560}% \special{pa 1186 1560}% \special{pa 1190 1560}% \special{pa 1196 1560}% \special{ip}% \special{pa 1196 1560}% \special{pa 1200 1560}% \special{pa 1206 1560}% \special{pa 1210 1560}% \special{pa 1216 1560}% \special{pa 1220 1560}% \special{pa 1226 1560}% \special{pa 1230 1560}% \special{pa 1236 1560}% \special{pa 1240 1560}% \special{pa 1246 1560}% \special{pa 1250 1560}% \special{pa 1256 1560}% \special{pa 1260 1560}% \special{pa 1266 1560}% \special{pa 1270 1560}% \special{pa 1276 1560}% \special{pa 1280 1560}% \special{pa 1286 1560}% \special{pa 1290 1560}% \special{pa 1296 1560}% \special{pa 1300 1560}% \special{pa 1306 1560}% \special{pa 1310 1560}% \special{pa 1316 1560}% \special{pa 1320 1560}% \special{pa 1326 1560}% \special{pa 1330 1560}% \special{pa 1336 1560}% \special{pa 1340 1560}% \special{pa 1346 1560}% \special{pa 1350 1560}% \special{pa 1356 1560}% \special{pa 1360 1560}% \special{pa 1366 1560}% \special{pa 1370 1560}% \special{pa 1376 1560}% \special{pa 1380 1560}% \special{pa 1386 1560}% \special{pa 1390 1560}% \special{pa 1396 1560}% \special{pa 1400 1560}% \special{pa 1406 1560}% \special{pa 1410 1560}% \special{pa 1416 1560}% \special{pa 1420 1560}% \special{pa 1426 1560}% \special{pa 1430 1560}% \special{pa 1436 1560}% \special{pa 1440 1560}% \special{pa 1446 1560}% \special{pa 1450 1560}% \special{pa 1456 1560}% \special{pa 1460 1560}% \special{pa 1466 1560}% \special{pa 1470 1560}% \special{pa 1476 1560}% \special{pa 1480 1560}% \special{pa 1486 1560}% \special{pa 1490 1560}% \special{pa 1496 1560}% \special{pa 1500 1560}% \special{pa 1506 1560}% \special{pa 1510 1560}% \special{pa 1516 1560}% \special{pa 1520 1560}% \special{pa 1526 1560}% \special{pa 1530 1560}% \special{pa 1536 1560}% \special{pa 1540 1560}% \special{pa 1546 1560}% \special{pa 1550 1560}% \special{pa 1556 1560}% \special{pa 1560 1560}% \special{pa 1566 1560}% \special{pa 1570 1560}% \special{pa 1576 1560}% \special{pa 1580 1560}% \special{pa 1586 1560}% \special{pa 1590 1560}% \special{pa 1596 1560}% \special{pa 1600 1560}% \special{pa 1606 1560}% \special{pa 1610 1560}% \special{pa 1616 1560}% \special{pa 1620 1560}% \special{pa 1626 1560}% \special{pa 1630 1560}% \special{pa 1636 1560}% \special{pa 1640 1560}% \special{pa 1646 1560}% \special{pa 1650 1560}% \special{pa 1656 1560}% \special{pa 1660 1560}% \special{pa 1666 1560}% \special{pa 1670 1560}% \special{pa 1676 1560}% \special{sp}% \special{pa 1680 1560}% \special{pa 1686 1560}% \special{pa 1690 1560}% \special{pa 1696 1560}% \special{pa 1700 1560}% \special{pa 1706 1560}% \special{pa 1710 1560}% \special{pa 1716 1560}% \special{pa 1720 1560}% \special{pa 1726 1560}% \special{pa 1730 1560}% \special{pa 1736 1560}% \special{pa 1740 1560}% \special{pa 1746 1560}% \special{pa 1750 1560}% \special{pa 1756 1560}% \special{pa 1760 1560}% \special{pa 1766 1560}% \special{pa 1770 1560}% \special{pa 1776 1560}% \special{pa 1780 1560}% \special{pa 1786 1560}% \special{pa 1790 1560}% \special{pa 1796 1560}% \special{pa 1800 1560}% \special{pa 1806 1560}% \special{pa 1810 1560}% \special{pa 1816 1560}% \special{pa 1820 1560}% \special{pa 1826 1560}% \special{pa 1830 1560}% \special{pa 1836 1560}% \special{pa 1840 1560}% \special{pa 1846 1560}% \special{pa 1850 1560}% \special{pa 1856 1560}% \special{pa 1860 1560}% \special{pa 1866 1560}% \special{pa 1870 1560}% \special{pa 1876 1560}% \special{pa 1880 1560}% \special{pa 1886 1560}% \special{pa 1890 1560}% \special{pa 1896 1560}% \special{pa 1900 1560}% \special{pa 1906 1560}% \special{pa 1910 1560}% \special{pa 1916 1560}% \special{pa 1920 1560}% \special{pa 1926 1560}% \special{pa 1930 1560}% \special{pa 1936 1560}% \special{pa 1940 1560}% \special{pa 1946 1560}% \special{pa 1950 1560}% \special{pa 1956 1560}% \special{pa 1960 1560}% \special{pa 1966 1560}% \special{pa 1970 1560}% \special{pa 1976 1560}% \special{pa 1980 1560}% \special{pa 1986 1560}% \special{pa 1990 1560}% \special{pa 1996 1560}% \special{pa 2000 1560}% \special{pa 2006 1560}% \special{pa 2010 1560}% \special{pa 2016 1560}% \special{pa 2020 1560}% \special{pa 2026 1560}% \special{pa 2030 1560}% \special{pa 2036 1560}% \special{pa 2040 1560}% \special{pa 2046 1560}% \special{pa 2050 1560}% \special{pa 2056 1560}% \special{pa 2060 1560}% \special{pa 2066 1560}% \special{pa 2070 1560}% \special{pa 2076 1560}% \special{pa 2080 1560}% \special{pa 2086 1560}% \special{pa 2090 1560}% \special{pa 2096 1560}% \special{pa 2100 1560}% \special{pa 2106 1560}% \special{pa 2110 1560}% \special{pa 2116 1560}% \special{pa 2120 1560}% \special{pa 2126 1560}% \special{pa 2130 1560}% \special{pa 2136 1560}% \special{pa 2140 1560}% \special{pa 2146 1560}% \special{pa 2150 1560}% \special{pa 2156 1560}% \special{pa 2160 1560}% \special{pa 2166 1560}% \special{pa 2170 1560}% \special{pa 2176 1560}% \special{pa 2180 1560}% \special{pa 2186 1560}% \special{pa 2190 1560}% \special{pa 2196 1560}% \special{pa 2200 1560}% \special{pa 2206 1560}% \special{pa 2210 1560}% \special{pa 2216 1560}% \special{pa 2220 1560}% \special{pa 2226 1560}% \special{pa 2230 1560}% \special{pa 2236 1560}% \special{pa 2240 1560}% \special{pa 2246 1560}% \special{pa 2250 1560}% \special{pa 2256 1560}% \special{pa 2260 1560}% \special{pa 2266 1560}% \special{pa 2270 1560}% \special{pa 2276 1560}% \special{pa 2280 1560}% \special{pa 2286 1560}% \special{pa 2290 1560}% \special{pa 2296 1560}% \special{pa 2300 1560}% \special{pa 2306 1560}% \special{pa 2310 1560}% \special{pa 2316 1560}% \special{pa 2320 1560}% \special{pa 2326 1560}% \special{pa 2330 1560}% \special{pa 2336 1560}% \special{pa 2340 1560}% \special{pa 2346 1560}% \special{pa 2350 1560}% \special{pa 2356 1560}% \special{pa 2360 1560}% \special{pa 2366 1560}% \special{pa 2370 1560}% \special{pa 2376 1560}% \special{pa 2380 1560}% \special{pa 2386 1560}% \special{pa 2390 1560}% \special{pa 2396 1560}% \special{pa 2400 1560}% \special{pa 2406 1560}% \special{pa 2410 1560}% \special{pa 2416 1560}% \special{pa 2420 1560}% \special{pa 2426 1560}% \special{pa 2430 1560}% \special{pa 2436 1560}% \special{pa 2440 1560}% \special{pa 2446 1560}% \special{pa 2450 1560}% \special{pa 2456 1560}% \special{pa 2460 1560}% \special{pa 2466 1560}% \special{pa 2470 1560}% \special{pa 2476 1560}% \special{pa 2480 1560}% \special{pa 2486 1560}% \special{pa 2490 1560}% \special{pa 2496 1560}% \special{pa 2500 1560}% \special{pa 2506 1560}% \special{pa 2510 1560}% \special{pa 2516 1560}% \special{pa 2520 1560}% \special{pa 2526 1560}% \special{pa 2530 1560}% \special{pa 2536 1560}% \special{pa 2540 1560}% \special{pa 2546 1560}% \special{pa 2550 1560}% \special{pa 2556 1560}% \special{pa 2560 1560}% \special{pa 2566 1560}% \special{pa 2570 1560}% \special{pa 2576 1560}% \special{pa 2580 1560}% \special{pa 2586 1560}% \special{pa 2590 1560}% \special{pa 2596 1560}% \special{pa 2600 1560}% \special{pa 2606 1560}% \special{pa 2610 1560}% \special{pa 2616 1560}% \special{pa 2620 1560}% \special{pa 2626 1560}% \special{pa 2630 1560}% \special{pa 2636 1560}% \special{pa 2640 1560}% \special{pa 2646 1560}% \special{pa 2650 1560}% \special{pa 2656 1560}% \special{pa 2660 1560}% \special{pa 2666 1560}% \special{pa 2670 1560}% \special{pa 2676 1560}% \special{pa 2680 1560}% \special{pa 2686 1560}% \special{pa 2690 1560}% \special{pa 2696 1560}% \special{pa 2700 1560}% \special{pa 2706 1560}% \special{pa 2710 1560}% \special{pa 2716 1560}% \special{pa 2720 1560}% \special{pa 2726 1560}% \special{pa 2730 1560}% \special{pa 2736 1560}% \special{pa 2740 1560}% \special{pa 2746 1560}% \special{pa 2750 1560}% \special{pa 2756 1560}% \special{pa 2760 1560}% \special{pa 2766 1560}% \special{pa 2770 1560}% \special{pa 2776 1560}% \special{pa 2780 1560}% \special{pa 2786 1560}% \special{pa 2790 1560}% \special{pa 2796 1560}% \special{pa 2800 1560}% \special{pa 2806 1560}% \special{pa 2810 1560}% \special{pa 2816 1560}% \special{pa 2820 1560}% \special{pa 2826 1560}% \special{pa 2830 1560}% \special{pa 2836 1560}% \special{pa 2840 1560}% \special{pa 2846 1560}% \special{pa 2850 1560}% \special{pa 2856 1560}% \special{pa 2860 1560}% \special{pa 2866 1560}% \special{pa 2870 1560}% \special{pa 2876 1560}% \special{pa 2880 1560}% \special{pa 2886 1560}% \special{pa 2890 1560}% \special{pa 2896 1560}% \special{pa 2900 1560}% \special{pa 2906 1560}% \special{pa 2910 1560}% \special{pa 2916 1560}% \special{pa 2920 1560}% \special{pa 2926 1560}% \special{pa 2930 1560}% \special{pa 2936 1560}% \special{pa 2940 1560}% \special{pa 2946 1560}% \special{pa 2950 1560}% \special{pa 2956 1560}% \special{pa 2960 1560}% \special{pa 2966 1560}% \special{pa 2970 1560}% \special{pa 2976 1560}% \special{pa 2980 1560}% \special{pa 2986 1560}% \special{pa 2990 1560}% \special{pa 2996 1560}% \special{pa 3000 1560}% \special{pa 3006 1560}% \special{pa 3010 1560}% \special{pa 3016 1560}% \special{pa 3020 1560}% \special{pa 3026 1560}% \special{pa 3030 1560}% \special{pa 3036 1560}% \special{pa 3040 1560}% \special{pa 3046 1560}% \special{pa 3050 1560}% \special{pa 3056 1560}% \special{pa 3060 1560}% \special{pa 3066 1560}% \special{pa 3070 1560}% \special{pa 3076 1560}% \special{pa 3080 1560}% \special{pa 3086 1560}% \special{pa 3090 1560}% \special{pa 3096 1560}% \special{pa 3100 1560}% \special{pa 3106 1560}% \special{pa 3110 1560}% \special{pa 3116 1560}% \special{pa 3120 1560}% \special{pa 3126 1560}% \special{pa 3130 1560}% \special{pa 3136 1560}% \special{pa 3140 1560}% \special{pa 3146 1560}% \special{pa 3150 1560}% \special{pa 3156 1560}% \special{pa 3160 1560}% \special{pa 3166 1560}% \special{pa 3170 1560}% \special{pa 3176 1560}% \special{pa 3180 1560}% \special{pa 3186 1560}% \special{pa 3190 1560}% \special{pa 3196 1560}% \special{pa 3200 1560}% \special{pa 3206 1560}% \special{pa 3210 1560}% \special{pa 3216 1560}% \special{pa 3220 1560}% \special{pa 3226 1560}% \special{pa 3230 1560}% \special{pa 3236 1560}% \special{pa 3240 1560}% \special{pa 3246 1560}% \special{pa 3250 1560}% \special{pa 3256 1560}% \special{pa 3260 1560}% \special{pa 3266 1560}% \special{pa 3270 1560}% \special{pa 3276 1560}% \special{pa 3280 1560}% \special{pa 3286 1560}% \special{pa 3290 1560}% \special{pa 3296 1560}% \special{pa 3300 1560}% \special{pa 3306 1560}% \special{pa 3310 1560}% \special{pa 3316 1560}% \special{pa 3320 1560}% \special{pa 3326 1560}% \special{pa 3330 1560}% \special{pa 3336 1560}% \special{pa 3340 1560}% \special{pa 3346 1560}% \special{pa 3350 1560}% \special{pa 3356 1560}% \special{ip}% % FUNC 1 0 3 0 % 10 958 1370 3358 2130 1198 2004 2158 2004 1198 1560 1678 1370 2158 2130 0 5 1 0 0 0 % sin(x) \special{pn 13}% \special{pa 1110 2130}% \special{pa 1116 2124}% \special{pa 1120 2116}% \special{pa 1126 2110}% \special{pa 1130 2102}% \special{pa 1136 2096}% \special{pa 1140 2088}% \special{pa 1146 2082}% \special{pa 1150 2074}% \special{pa 1156 2066}% \special{pa 1160 2060}% \special{pa 1166 2052}% \special{pa 1170 2046}% \special{pa 1176 2038}% \special{pa 1180 2030}% \special{pa 1186 2024}% \special{pa 1190 2016}% \special{pa 1196 2008}% \special{pa 1200 2002}% \special{pa 1206 1994}% \special{pa 1210 1988}% \special{pa 1216 1980}% \special{pa 1220 1972}% \special{pa 1226 1966}% \special{pa 1230 1958}% \special{pa 1236 1950}% \special{pa 1240 1944}% \special{pa 1246 1936}% \special{pa 1250 1930}% \special{pa 1256 1922}% \special{pa 1260 1916}% \special{pa 1266 1908}% \special{pa 1270 1900}% \special{pa 1276 1894}% \special{pa 1280 1886}% \special{pa 1286 1880}% \special{pa 1290 1872}% \special{pa 1296 1866}% \special{pa 1300 1860}% \special{pa 1306 1852}% \special{pa 1310 1846}% \special{pa 1316 1838}% \special{pa 1320 1832}% \special{pa 1326 1826}% \special{pa 1330 1818}% \special{pa 1336 1812}% \special{pa 1340 1806}% \special{pa 1346 1800}% \special{pa 1350 1792}% \special{pa 1356 1786}% \special{pa 1360 1780}% \special{pa 1366 1774}% \special{pa 1370 1768}% \special{pa 1376 1762}% \special{pa 1380 1756}% \special{pa 1386 1750}% \special{pa 1390 1744}% \special{pa 1396 1738}% \special{pa 1400 1732}% \special{pa 1406 1726}% \special{pa 1410 1720}% \special{pa 1416 1716}% \special{pa 1420 1710}% \special{pa 1426 1704}% \special{pa 1430 1698}% \special{pa 1436 1694}% \special{pa 1440 1688}% \special{pa 1446 1684}% \special{pa 1450 1678}% \special{pa 1456 1674}% \special{pa 1460 1668}% \special{pa 1466 1664}% \special{pa 1470 1660}% \special{pa 1476 1654}% \special{pa 1480 1650}% \special{pa 1486 1646}% \special{pa 1490 1642}% \special{pa 1496 1638}% \special{pa 1500 1634}% \special{pa 1506 1630}% \special{pa 1510 1626}% \special{pa 1516 1622}% \special{pa 1520 1618}% \special{pa 1526 1616}% \special{pa 1530 1612}% \special{pa 1536 1608}% \special{pa 1540 1606}% \special{pa 1546 1602}% \special{pa 1550 1598}% \special{pa 1556 1596}% \special{pa 1560 1594}% \special{pa 1566 1590}% \special{pa 1570 1588}% \special{pa 1576 1586}% \special{pa 1580 1584}% \special{pa 1586 1580}% \special{pa 1590 1578}% \special{pa 1596 1576}% \special{pa 1600 1574}% \special{pa 1606 1574}% \special{pa 1610 1572}% \special{pa 1616 1570}% \special{pa 1620 1568}% \special{pa 1626 1568}% \special{pa 1630 1566}% \special{pa 1636 1564}% \special{pa 1640 1564}% \special{pa 1646 1564}% \special{pa 1650 1562}% \special{pa 1656 1562}% \special{pa 1660 1562}% \special{pa 1666 1560}% \special{pa 1670 1560}% \special{pa 1676 1560}% \special{ip}% \special{pa 1676 1560}% \special{pa 1680 1560}% \special{pa 1686 1560}% \special{pa 1690 1560}% \special{pa 1696 1562}% \special{pa 1700 1562}% \special{pa 1706 1562}% \special{pa 1710 1562}% \special{pa 1716 1564}% \special{pa 1720 1564}% \special{pa 1726 1566}% \special{pa 1730 1566}% \special{pa 1736 1568}% \special{pa 1740 1570}% \special{pa 1746 1572}% \special{pa 1750 1572}% \special{pa 1756 1574}% \special{pa 1760 1576}% \special{pa 1766 1578}% \special{pa 1770 1580}% \special{pa 1776 1582}% \special{pa 1780 1586}% \special{pa 1786 1588}% \special{pa 1790 1590}% \special{pa 1796 1592}% \special{pa 1800 1596}% \special{pa 1806 1598}% \special{pa 1810 1602}% \special{pa 1816 1604}% \special{pa 1820 1608}% \special{pa 1826 1610}% \special{pa 1830 1614}% \special{pa 1836 1618}% \special{pa 1840 1622}% \special{pa 1846 1626}% \special{pa 1850 1628}% \special{pa 1856 1632}% \special{pa 1860 1636}% \special{pa 1866 1642}% \special{pa 1870 1646}% \special{pa 1876 1650}% \special{pa 1880 1654}% \special{pa 1886 1658}% \special{pa 1890 1664}% \special{pa 1896 1668}% \special{pa 1900 1672}% \special{pa 1906 1678}% \special{pa 1910 1682}% \special{pa 1916 1688}% \special{pa 1920 1692}% \special{pa 1926 1698}% \special{pa 1930 1704}% \special{pa 1936 1708}% \special{pa 1940 1714}% \special{pa 1946 1720}% \special{pa 1950 1726}% \special{pa 1956 1730}% \special{pa 1960 1736}% \special{pa 1966 1742}% \special{pa 1970 1748}% \special{pa 1976 1754}% \special{pa 1980 1760}% \special{pa 1986 1766}% \special{pa 1990 1772}% \special{pa 1996 1778}% \special{pa 2000 1786}% \special{pa 2006 1792}% \special{pa 2010 1798}% \special{pa 2016 1804}% \special{pa 2020 1810}% \special{pa 2026 1818}% \special{pa 2030 1824}% \special{pa 2036 1830}% \special{pa 2040 1838}% \special{pa 2046 1844}% \special{pa 2050 1850}% \special{pa 2056 1858}% \special{pa 2060 1864}% \special{pa 2066 1872}% \special{pa 2070 1878}% \special{pa 2076 1886}% \special{pa 2080 1892}% \special{pa 2086 1900}% \special{pa 2090 1906}% \special{pa 2096 1914}% \special{pa 2100 1920}% \special{pa 2106 1928}% \special{pa 2110 1936}% \special{pa 2116 1942}% \special{pa 2120 1950}% \special{pa 2126 1956}% \special{pa 2130 1964}% \special{pa 2136 1972}% \special{pa 2140 1978}% \special{pa 2146 1986}% \special{pa 2150 1992}% \special{pa 2156 2000}% \special{sp}% \special{pa 2160 2008}% \special{pa 2166 2014}% \special{pa 2170 2022}% \special{pa 2176 2030}% \special{pa 2180 2036}% \special{pa 2186 2044}% \special{pa 2190 2050}% \special{pa 2196 2058}% \special{pa 2200 2066}% \special{pa 2206 2072}% \special{pa 2210 2080}% \special{pa 2216 2086}% \special{pa 2220 2094}% \special{pa 2226 2102}% \special{pa 2230 2108}% \special{pa 2236 2116}% \special{pa 2240 2122}% \special{pa 2246 2130}% \special{pa 2246 2130}% \special{ip}% \special{pa 3030 2130}% \special{pa 3036 2124}% \special{pa 3040 2116}% \special{pa 3046 2110}% \special{pa 3050 2102}% \special{pa 3056 2096}% \special{pa 3060 2088}% \special{pa 3066 2082}% \special{pa 3070 2074}% \special{pa 3076 2066}% \special{pa 3080 2060}% \special{pa 3086 2052}% \special{pa 3090 2046}% \special{pa 3096 2038}% \special{pa 3100 2030}% \special{pa 3106 2024}% \special{pa 3110 2016}% \special{pa 3116 2008}% \special{pa 3120 2002}% \special{pa 3126 1994}% \special{pa 3130 1988}% \special{pa 3136 1980}% \special{pa 3140 1972}% \special{pa 3146 1966}% \special{pa 3150 1958}% \special{pa 3156 1950}% \special{pa 3160 1944}% \special{pa 3166 1936}% \special{pa 3170 1930}% \special{pa 3176 1922}% \special{pa 3180 1916}% \special{pa 3186 1908}% \special{pa 3190 1900}% \special{pa 3196 1894}% \special{pa 3200 1886}% \special{pa 3206 1880}% \special{pa 3210 1872}% \special{pa 3216 1866}% \special{pa 3220 1860}% \special{pa 3226 1852}% \special{pa 3230 1846}% \special{pa 3236 1838}% \special{pa 3240 1832}% \special{pa 3246 1826}% \special{pa 3250 1818}% \special{pa 3256 1812}% \special{pa 3260 1806}% \special{pa 3266 1800}% \special{pa 3270 1792}% \special{pa 3276 1786}% \special{pa 3280 1780}% \special{pa 3286 1774}% \special{pa 3290 1768}% \special{pa 3296 1762}% \special{pa 3300 1756}% \special{pa 3306 1750}% \special{pa 3310 1744}% \special{pa 3316 1738}% \special{pa 3320 1732}% \special{pa 3326 1726}% \special{pa 3330 1720}% \special{pa 3336 1716}% \special{pa 3340 1710}% \special{pa 3346 1704}% \special{pa 3350 1698}% \special{pa 3356 1694}% \special{ip}% % FUNC 1 0 3 0 % 10 960 1370 3360 2130 1200 2004 2160 2004 1200 1560 2640 1370 3120 2130 0 5 1 0 0 0 % cos(x) \special{pn 13}% \special{pa 960 1690}% \special{pa 966 1686}% \special{pa 970 1680}% \special{pa 976 1676}% \special{pa 980 1670}% \special{pa 986 1666}% \special{pa 990 1662}% \special{pa 996 1656}% \special{pa 1000 1652}% \special{pa 1006 1648}% \special{pa 1010 1644}% \special{pa 1016 1640}% \special{pa 1020 1636}% \special{pa 1026 1632}% \special{pa 1030 1628}% \special{pa 1036 1624}% \special{pa 1040 1620}% \special{pa 1046 1616}% \special{pa 1050 1612}% \special{pa 1056 1610}% \special{pa 1060 1606}% \special{pa 1066 1604}% \special{pa 1070 1600}% \special{pa 1076 1598}% \special{pa 1080 1594}% \special{pa 1086 1592}% \special{pa 1090 1588}% \special{pa 1096 1586}% \special{pa 1100 1584}% \special{pa 1106 1582}% \special{pa 1110 1580}% \special{pa 1116 1578}% \special{pa 1120 1576}% \special{pa 1126 1574}% \special{pa 1130 1572}% \special{pa 1136 1570}% \special{pa 1140 1570}% \special{pa 1146 1568}% \special{pa 1150 1566}% \special{pa 1156 1566}% \special{pa 1160 1564}% \special{pa 1166 1564}% \special{pa 1170 1562}% \special{pa 1176 1562}% \special{pa 1180 1562}% \special{pa 1186 1562}% \special{pa 1190 1560}% \special{pa 1196 1560}% \special{pa 1200 1560}% \special{pa 1206 1560}% \special{pa 1210 1560}% \special{pa 1216 1562}% \special{pa 1220 1562}% \special{pa 1226 1562}% \special{pa 1230 1562}% \special{pa 1236 1564}% \special{pa 1240 1564}% \special{pa 1246 1566}% \special{pa 1250 1566}% \special{pa 1256 1568}% \special{pa 1260 1570}% \special{pa 1266 1570}% \special{pa 1270 1572}% \special{pa 1276 1574}% \special{pa 1280 1576}% \special{pa 1286 1578}% \special{pa 1290 1580}% \special{pa 1296 1582}% \special{pa 1300 1584}% \special{pa 1306 1586}% \special{pa 1310 1588}% \special{pa 1316 1592}% \special{pa 1320 1594}% \special{pa 1326 1598}% \special{pa 1330 1600}% \special{pa 1336 1604}% \special{pa 1340 1606}% \special{pa 1346 1610}% \special{pa 1350 1612}% \special{pa 1356 1616}% \special{pa 1360 1620}% \special{pa 1366 1624}% \special{pa 1370 1628}% \special{pa 1376 1632}% \special{pa 1380 1636}% \special{pa 1386 1640}% \special{pa 1390 1644}% \special{pa 1396 1648}% \special{pa 1400 1652}% \special{pa 1406 1656}% \special{pa 1410 1662}% \special{pa 1416 1666}% \special{pa 1420 1670}% \special{pa 1426 1676}% \special{pa 1430 1680}% \special{pa 1436 1686}% \special{pa 1440 1690}% \special{pa 1446 1696}% \special{pa 1450 1700}% \special{pa 1456 1706}% \special{pa 1460 1712}% \special{pa 1466 1718}% \special{pa 1470 1722}% \special{pa 1476 1728}% \special{pa 1480 1734}% \special{pa 1486 1740}% \special{pa 1490 1746}% \special{pa 1496 1752}% \special{pa 1500 1758}% \special{pa 1506 1764}% \special{pa 1510 1770}% \special{pa 1516 1776}% \special{pa 1520 1782}% \special{pa 1526 1788}% \special{pa 1530 1796}% \special{pa 1536 1802}% \special{pa 1540 1808}% \special{pa 1546 1814}% \special{pa 1550 1822}% \special{pa 1556 1828}% \special{pa 1560 1834}% \special{pa 1566 1842}% \special{pa 1570 1848}% \special{pa 1576 1854}% \special{pa 1580 1862}% \special{pa 1586 1868}% \special{pa 1590 1876}% \special{pa 1596 1882}% \special{pa 1600 1890}% \special{pa 1606 1896}% \special{pa 1610 1904}% \special{pa 1616 1910}% \special{pa 1620 1918}% \special{pa 1626 1926}% \special{pa 1630 1932}% \special{pa 1636 1940}% \special{pa 1640 1946}% \special{pa 1646 1954}% \special{pa 1650 1960}% \special{pa 1656 1968}% \special{pa 1660 1976}% \special{pa 1666 1982}% \special{pa 1670 1990}% \special{pa 1676 1998}% \special{pa 1680 2004}% \special{pa 1686 2012}% \special{pa 1690 2020}% \special{pa 1696 2026}% \special{pa 1700 2034}% \special{pa 1706 2040}% \special{pa 1710 2048}% \special{pa 1716 2056}% \special{pa 1720 2062}% \special{pa 1726 2070}% \special{pa 1730 2076}% \special{pa 1736 2084}% \special{pa 1740 2092}% \special{pa 1746 2098}% \special{pa 1750 2106}% \special{pa 1756 2112}% \special{pa 1760 2120}% \special{pa 1766 2126}% \special{pa 1768 2130}% \special{ip}% \special{pa 2552 2130}% \special{pa 2556 2126}% \special{pa 2560 2120}% \special{pa 2566 2112}% \special{pa 2570 2106}% \special{pa 2576 2098}% \special{pa 2580 2092}% \special{pa 2586 2084}% \special{pa 2590 2076}% \special{pa 2596 2070}% \special{pa 2600 2062}% \special{pa 2606 2056}% \special{pa 2610 2048}% \special{pa 2616 2040}% \special{pa 2620 2034}% \special{pa 2626 2026}% \special{pa 2630 2020}% \special{pa 2636 2012}% \special{ip}% \special{pa 2640 2004}% \special{pa 2646 1998}% \special{pa 2650 1990}% \special{pa 2656 1982}% \special{pa 2660 1976}% \special{pa 2666 1968}% \special{pa 2670 1960}% \special{pa 2676 1954}% \special{pa 2680 1946}% \special{pa 2686 1940}% \special{pa 2690 1932}% \special{pa 2696 1926}% \special{pa 2700 1918}% \special{pa 2706 1910}% \special{pa 2710 1904}% \special{pa 2716 1896}% \special{pa 2720 1890}% \special{pa 2726 1882}% \special{pa 2730 1876}% \special{pa 2736 1868}% \special{pa 2740 1862}% \special{pa 2746 1854}% \special{pa 2750 1848}% \special{pa 2756 1842}% \special{pa 2760 1834}% \special{pa 2766 1828}% \special{pa 2770 1822}% \special{pa 2776 1814}% \special{pa 2780 1808}% \special{pa 2786 1802}% \special{pa 2790 1796}% \special{pa 2796 1788}% \special{pa 2800 1782}% \special{pa 2806 1776}% \special{pa 2810 1770}% \special{pa 2816 1764}% \special{pa 2820 1758}% \special{pa 2826 1752}% \special{pa 2830 1746}% \special{pa 2836 1740}% \special{pa 2840 1734}% \special{pa 2846 1728}% \special{pa 2850 1722}% \special{pa 2856 1718}% \special{pa 2860 1712}% \special{pa 2866 1706}% \special{pa 2870 1700}% \special{pa 2876 1696}% \special{pa 2880 1690}% \special{pa 2886 1686}% \special{pa 2890 1680}% \special{pa 2896 1676}% \special{pa 2900 1670}% \special{pa 2906 1666}% \special{pa 2910 1662}% \special{pa 2916 1656}% \special{pa 2920 1652}% \special{pa 2926 1648}% \special{pa 2930 1644}% \special{pa 2936 1640}% \special{pa 2940 1636}% \special{pa 2946 1632}% \special{pa 2950 1628}% \special{pa 2956 1624}% \special{pa 2960 1620}% \special{pa 2966 1616}% \special{pa 2970 1612}% \special{pa 2976 1610}% \special{pa 2980 1606}% \special{pa 2986 1604}% \special{pa 2990 1600}% \special{pa 2996 1598}% \special{pa 3000 1594}% \special{pa 3006 1592}% \special{pa 3010 1588}% \special{pa 3016 1586}% \special{pa 3020 1584}% \special{pa 3026 1582}% \special{pa 3030 1580}% \special{pa 3036 1578}% \special{pa 3040 1576}% \special{pa 3046 1574}% \special{pa 3050 1572}% \special{pa 3056 1570}% \special{pa 3060 1570}% \special{pa 3066 1568}% \special{pa 3070 1566}% \special{pa 3076 1566}% \special{pa 3080 1564}% \special{pa 3086 1564}% \special{pa 3090 1562}% \special{pa 3096 1562}% \special{pa 3100 1562}% \special{pa 3106 1562}% \special{pa 3110 1560}% \special{pa 3116 1560}% \special{pa 3120 1560}% \special{sp}% \special{pa 3120 1560}% \special{pa 3126 1560}% \special{pa 3130 1560}% \special{pa 3136 1562}% \special{pa 3140 1562}% \special{pa 3146 1562}% \special{pa 3150 1562}% \special{pa 3156 1564}% \special{pa 3160 1564}% \special{pa 3166 1566}% \special{pa 3170 1566}% \special{pa 3176 1568}% \special{pa 3180 1570}% \special{pa 3186 1570}% \special{pa 3190 1572}% \special{pa 3196 1574}% \special{pa 3200 1576}% \special{pa 3206 1578}% \special{pa 3210 1580}% \special{pa 3216 1582}% \special{pa 3220 1584}% \special{pa 3226 1586}% \special{pa 3230 1588}% \special{pa 3236 1592}% \special{pa 3240 1594}% \special{pa 3246 1598}% \special{pa 3250 1600}% \special{pa 3256 1604}% \special{pa 3260 1606}% \special{pa 3266 1610}% \special{pa 3270 1612}% \special{pa 3276 1616}% \special{pa 3280 1620}% \special{pa 3286 1624}% \special{pa 3290 1628}% \special{pa 3296 1632}% \special{pa 3300 1636}% \special{pa 3306 1640}% \special{pa 3310 1644}% \special{pa 3316 1648}% \special{pa 3320 1652}% \special{pa 3326 1656}% \special{pa 3330 1662}% \special{pa 3336 1666}% \special{pa 3340 1670}% \special{pa 3346 1676}% \special{pa 3350 1680}% \special{pa 3356 1686}% \special{pa 3360 1690}% \special{ip}% % LINE 1 0 3 0 % 2 2150 2000 2660 2000 % \special{pn 13}% \special{pa 2150 2000}% \special{pa 2660 2000}% \special{fp}% % STR 2 0 3 0 % 3 1070 2030 1070 2130 2 0 % {\small O} \put(10.7000,-21.3000){\makebox(0,0)[lb]{{\small O}}}% % LINE 3 0 3 0 % 2 1680 2000 1680 1560 % \special{pn 4}% \special{pa 1680 2000}% \special{pa 1680 1560}% \special{fp}% % LINE 3 0 3 0 % 2 3120 2000 3120 1560 % \special{pn 4}% \special{pa 3120 2000}% \special{pa 3120 1560}% \special{fp}% % STR 2 0 3 0 % 3 1230 1420 1230 1520 2 0 % {\small 1} \put(12.3000,-15.2000){\makebox(0,0)[lb]{{\small 1}}}% % STR 2 0 3 0 % 3 1630 2110 1630 2210 2 0 % $\frac{\pi}{2}$ \put(16.3000,-22.1000){\makebox(0,0)[lb]{$\frac{\pi}{2}$}}% % STR 2 0 3 0 % 3 2570 2140 2570 2240 2 0 % $\frac{3\pi}{2}$ \put(25.7000,-22.4000){\makebox(0,0)[lb]{$\frac{3\pi}{2}$}}% % STR 2 0 3 0 % 3 2100 2020 2100 2120 2 0 % $\pi$ \put(21.0000,-21.2000){\makebox(0,0)[lb]{$\pi$}}% % STR 2 0 3 0 % 3 3030 2040 3030 2140 2 0 % $2\pi$ \put(30.3000,-21.4000){\makebox(0,0)[lb]{$2\pi$}}% % STR 2 0 3 0 % 3 3400 1950 3400 2050 2 0 % $\theta$ \put(34.0000,-20.5000){\makebox(0,0)[lb]{$\theta$}}% \end{picture}% \end{center} \end{enumerate} \end{document}