津田塾大学 国際関係学科 2003年度 問2

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解答作成者: 森 宏征

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入試情報

大学名 津田塾大学
学科・方式 国際関係学科
年度 2003年度
問No 問2
学部 国際関係学科
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,dvipdfmx]{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} \def\Op{{\mathrm{O}}} \def\LLL{{\lll}} \def\RRR{{\rrr}} \def\LL{{\ll}} \def\RR{{\rr}} \def\cdotts{{\cdots\cdotssp}} \usepackage{graphicx} \usepackage{pifont} \usepackage{fancybox} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  $0^\circ \leqq x < 180^\circ\enskip(x \neq 90^\circ),\enskip 0^\circ \leqq y < 90^\circ,\enskip -1 \leqq \tan x \tan y \leqq 1$ をみたす$(x,\ y)$の範囲を図示せよ. \end{FRAME} \vskip 2mm \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \vspace{2mm} $-1 \leqq \tan x\tan y \leqq 1$ より, \begin{align*} -1 \leqq \frac{\sin x\sin y}{\cos x\cos y} \leqq 1 \tag*{$\cdott\MARU{1}$} \end{align*} \MARU{1}をみたす$(x,\ y)$の集合 $E$ が求めるものである. \vspace{-2mm} \begin{enumerate} \renewcommand{\labelenumi}{(\roman{enumi})} \item $0^\circ \leqq x < 90^\circ$ のとき  $\cos x\cos y > 0,\enskip \sin x\sin y \geqq 0$ だから\MARU{1}より, \begin{gather*} -\cos x\cos y \leqq \sin x \sin y \leqq \cos x \cos y \\ \qquad \cos x\cos y + \sin x\sin y \geqq 0 \enskip \wedge \enskip \cos x\cos y - \sin x\sin y \geqq 0 \\ \qquad \cos(x + y) \geqq 0 \quad (\,\because\enskip \cos x\cos y + \sin x\sin y \geqq 0) \\ \therefore \enskip 0^\circ \leqq x + y \leqq 90^\circ \tag*{$\cdott\MARU{2}$} \end{gather*} \item $90^\circ < x < 180^\circ$ のとき  $\cos x \cos y < 0,\enskip \sin x\sin y > 0$ だから\MARU{1}より, \begin{gather*} -\cos x\cos y \geqq \sin x\sin y \geqq \cos x\cos y \\ \qquad \cos x \cos y + \sin x\sin y \leqq 0 \enskip \wedge \enskip \cos x\cos y - \sin x\sin y \leqq 0 \\ \qquad \cos(x - y) \leqq 0 \quad (\,\because\enskip \cos x\cos y- \sin x\sin y < 0) \\ \therefore \enskip 90^\circ < x - y < 180^\circ \tag*{$\cdott\MARU{3}$} \end{gather*} \end{enumerate} (i),\enskip(ii)より $E$ は$(\MARU{2}\vee\MARU{3})$に よって定まる集合である. これを図示すると下図の下線部分の通り. ただし,境界は白丸および点線部分は除く. \begin{center} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 21.4000, 15.1800)( 8.5000,-22.5800) % VECTOR 2 0 3 0 % 2 903 2016 2948 2016 % \special{pn 8}% \special{pa 904 2016}% \special{pa 2948 2016}% \special{fp}% \special{sh 1}% \special{pa 2948 2016}% \special{pa 2882 1996}% \special{pa 2896 2016}% \special{pa 2882 2036}% \special{pa 2948 2016}% \special{fp}% % VECTOR 2 0 3 0 % 2 1145 2258 1145 806 % \special{pn 8}% \special{pa 1146 2258}% \special{pa 1146 806}% \special{fp}% \special{sh 1}% \special{pa 1146 806}% \special{pa 1126 874}% \special{pa 1146 860}% \special{pa 1166 874}% \special{pa 1146 806}% \special{fp}% % LINE 0 2 3 0 % 2 2597 1290 2597 2016 % \special{pn 20}% \special{pa 2598 1290}% \special{pa 2598 2016}% \special{dt 0.054}% % LINE 2 2 3 0 % 2 2597 1290 1145 1290 % \special{pn 8}% \special{pa 2598 1290}% \special{pa 1146 1290}% \special{dt 0.045}% % CIRCLE 2 0 3 0 % 4 2597 1290 2597 1338 2597 1338 2597 1338 % \special{pn 8}% \special{ar 2598 1290 48 48 0.0000000 6.2831853}% % CIRCLE 2 0 3 0 % 4 1145 1290 1145 1338 1145 1338 1145 1338 % \special{pn 8}% \special{ar 1146 1290 48 48 0.0000000 6.2831853}% % CIRCLE 2 0 3 0 % 4 1871 2016 1871 2064 1871 2064 1871 2064 % \special{pn 8}% \special{ar 1872 2016 48 48 0.0000000 6.2831853}% % CIRCLE 2 0 3 0 % 4 2597 2016 2597 2064 2597 2064 2597 2064 % \special{pn 8}% \special{ar 2598 2016 48 48 0.0000000 6.2831853}% % STR 2 0 3 0 % 3 1200 789 1200 910 2 0 % $y$ \put(12.0000,-9.1000){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 2990 1919 2990 2040 2 0 % $x$ \put(29.9000,-20.4000){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 850 1199 850 1320 2 0 % {\footnotesize$90^\circ$} \put(8.5000,-13.2000){\makebox(0,0)[lb]{{\footnotesize$90^\circ$}}}% % STR 2 0 3 0 % 3 1770 2099 1770 2220 2 0 % {\footnotesize$90^\circ$} \put(17.7000,-22.2000){\makebox(0,0)[lb]{{\footnotesize$90^\circ$}}}% % STR 2 0 3 0 % 3 2460 2099 2460 2220 2 0 % {\footnotesize$180^\circ$} \put(24.6000,-22.2000){\makebox(0,0)[lb]{{\footnotesize$180^\circ$}}}% % STR 2 0 3 0 % 3 1020 2040 1020 2150 2 0 % {\small O} \put(10.2000,-21.5000){\makebox(0,0)[lb]{{\small O}}}% % LINE 0 0 3 0 % 2 1177 1323 1837 1983 % \special{pn 20}% \special{pa 1178 1324}% \special{pa 1838 1984}% \special{fp}% % LINE 0 0 3 0 % 2 1903 1983 2563 1323 % \special{pn 20}% \special{pa 1904 1984}% \special{pa 2564 1324}% \special{fp}% % LINE 3 0 3 0 % 60 1446 1604 1144 1906 1463 1620 1144 1939 1479 1637 1144 1972 1496 1653 1144 2005 1512 1670 1166 2016 1529 1686 1199 2016 1545 1703 1232 2016 1562 1719 1265 2016 1578 1736 1298 2016 1595 1752 1331 2016 1611 1769 1364 2016 1628 1785 1397 2016 1644 1802 1430 2016 1661 1818 1463 2016 1677 1835 1496 2016 1694 1851 1529 2016 1710 1868 1562 2016 1727 1884 1595 2016 1743 1901 1628 2016 1760 1917 1661 2016 1776 1934 1694 2016 1793 1950 1727 2016 1809 1967 1760 2016 1826 1983 1793 2016 1430 1587 1144 1873 1413 1571 1144 1840 1397 1554 1144 1807 1380 1538 1144 1774 1364 1521 1144 1741 1347 1505 1144 1708 % \special{pn 4}% \special{pa 1446 1604}% \special{pa 1144 1906}% \special{fp}% \special{pa 1464 1620}% \special{pa 1144 1940}% \special{fp}% \special{pa 1480 1638}% \special{pa 1144 1972}% \special{fp}% \special{pa 1496 1654}% \special{pa 1144 2006}% \special{fp}% \special{pa 1512 1670}% \special{pa 1166 2016}% \special{fp}% \special{pa 1530 1686}% \special{pa 1200 2016}% \special{fp}% \special{pa 1546 1704}% \special{pa 1232 2016}% \special{fp}% \special{pa 1562 1720}% \special{pa 1266 2016}% \special{fp}% \special{pa 1578 1736}% \special{pa 1298 2016}% \special{fp}% \special{pa 1596 1752}% \special{pa 1332 2016}% \special{fp}% \special{pa 1612 1770}% \special{pa 1364 2016}% \special{fp}% \special{pa 1628 1786}% \special{pa 1398 2016}% \special{fp}% \special{pa 1644 1802}% \special{pa 1430 2016}% \special{fp}% \special{pa 1662 1818}% \special{pa 1464 2016}% \special{fp}% \special{pa 1678 1836}% \special{pa 1496 2016}% \special{fp}% \special{pa 1694 1852}% \special{pa 1530 2016}% \special{fp}% \special{pa 1710 1868}% \special{pa 1562 2016}% \special{fp}% \special{pa 1728 1884}% \special{pa 1596 2016}% \special{fp}% \special{pa 1744 1902}% \special{pa 1628 2016}% \special{fp}% \special{pa 1760 1918}% \special{pa 1662 2016}% \special{fp}% \special{pa 1776 1934}% \special{pa 1694 2016}% \special{fp}% \special{pa 1794 1950}% \special{pa 1728 2016}% \special{fp}% \special{pa 1810 1968}% \special{pa 1760 2016}% \special{fp}% \special{pa 1826 1984}% \special{pa 1794 2016}% \special{fp}% \special{pa 1430 1588}% \special{pa 1144 1874}% \special{fp}% \special{pa 1414 1572}% \special{pa 1144 1840}% \special{fp}% \special{pa 1398 1554}% \special{pa 1144 1808}% \special{fp}% \special{pa 1380 1538}% \special{pa 1144 1774}% \special{fp}% \special{pa 1364 1522}% \special{pa 1144 1742}% \special{fp}% \special{pa 1348 1506}% \special{pa 1144 1708}% \special{fp}% % LINE 3 0 3 1 % 20 1331 1488 1144 1675 1314 1472 1144 1642 1298 1455 1144 1609 1281 1439 1144 1576 1265 1422 1144 1543 1248 1406 1144 1510 1232 1389 1144 1477 1215 1373 1144 1444 1199 1356 1144 1411 1182 1340 1144 1378 % \special{pn 4}% \special{pa 1332 1488}% \special{pa 1144 1676}% \special{fp}% \special{pa 1314 1472}% \special{pa 1144 1642}% \special{fp}% \special{pa 1298 1456}% \special{pa 1144 1610}% \special{fp}% \special{pa 1282 1440}% \special{pa 1144 1576}% \special{fp}% \special{pa 1266 1422}% \special{pa 1144 1544}% \special{fp}% \special{pa 1248 1406}% \special{pa 1144 1510}% \special{fp}% \special{pa 1232 1390}% \special{pa 1144 1478}% \special{fp}% \special{pa 1216 1374}% \special{pa 1144 1444}% \special{fp}% \special{pa 1200 1356}% \special{pa 1144 1412}% \special{fp}% \special{pa 1182 1340}% \special{pa 1144 1378}% \special{fp}% % LINE 3 0 3 0 % 60 2596 1895 2299 1598 2596 1862 2315 1582 2596 1829 2332 1565 2596 1796 2348 1549 2596 1763 2365 1532 2596 1730 2381 1516 2596 1697 2398 1499 2596 1664 2414 1483 2596 1631 2431 1466 2596 1598 2447 1450 2596 1565 2464 1433 2596 1532 2480 1417 2596 1499 2497 1400 2596 1466 2513 1384 2596 1433 2530 1367 2596 1400 2546 1351 2596 1367 2563 1334 2596 1928 2282 1615 2596 1961 2266 1631 2574 1972 2249 1648 2557 1989 2233 1664 2546 2011 2216 1681 2519 2016 2200 1697 2486 2016 2183 1714 2453 2016 2167 1730 2420 2016 2150 1747 2387 2016 2134 1763 2354 2016 2117 1780 2321 2016 2101 1796 2288 2016 2084 1813 % \special{pn 4}% \special{pa 2596 1896}% \special{pa 2300 1598}% \special{fp}% \special{pa 2596 1862}% \special{pa 2316 1582}% \special{fp}% \special{pa 2596 1830}% \special{pa 2332 1566}% \special{fp}% \special{pa 2596 1796}% \special{pa 2348 1550}% \special{fp}% \special{pa 2596 1764}% \special{pa 2366 1532}% \special{fp}% \special{pa 2596 1730}% \special{pa 2382 1516}% \special{fp}% \special{pa 2596 1698}% \special{pa 2398 1500}% \special{fp}% \special{pa 2596 1664}% \special{pa 2414 1484}% \special{fp}% \special{pa 2596 1632}% \special{pa 2432 1466}% \special{fp}% \special{pa 2596 1598}% \special{pa 2448 1450}% \special{fp}% \special{pa 2596 1566}% \special{pa 2464 1434}% \special{fp}% \special{pa 2596 1532}% \special{pa 2480 1418}% \special{fp}% \special{pa 2596 1500}% \special{pa 2498 1400}% \special{fp}% \special{pa 2596 1466}% \special{pa 2514 1384}% \special{fp}% \special{pa 2596 1434}% \special{pa 2530 1368}% \special{fp}% \special{pa 2596 1400}% \special{pa 2546 1352}% \special{fp}% \special{pa 2596 1368}% \special{pa 2564 1334}% \special{fp}% \special{pa 2596 1928}% \special{pa 2282 1616}% \special{fp}% \special{pa 2596 1962}% \special{pa 2266 1632}% \special{fp}% \special{pa 2574 1972}% \special{pa 2250 1648}% \special{fp}% \special{pa 2558 1990}% \special{pa 2234 1664}% \special{fp}% \special{pa 2546 2012}% \special{pa 2216 1682}% \special{fp}% \special{pa 2520 2016}% \special{pa 2200 1698}% \special{fp}% \special{pa 2486 2016}% \special{pa 2184 1714}% \special{fp}% \special{pa 2454 2016}% \special{pa 2168 1730}% \special{fp}% \special{pa 2420 2016}% \special{pa 2150 1748}% \special{fp}% \special{pa 2388 2016}% \special{pa 2134 1764}% \special{fp}% \special{pa 2354 2016}% \special{pa 2118 1780}% \special{fp}% \special{pa 2322 2016}% \special{pa 2102 1796}% \special{fp}% \special{pa 2288 2016}% \special{pa 2084 1814}% \special{fp}% % LINE 3 0 3 1 % 20 2255 2016 2068 1829 2222 2016 2051 1846 2189 2016 2035 1862 2156 2016 2018 1879 2123 2016 2002 1895 2090 2016 1985 1912 2057 2016 1969 1928 2024 2016 1952 1945 1991 2016 1936 1961 1958 2016 1919 1978 % \special{pn 4}% \special{pa 2256 2016}% \special{pa 2068 1830}% \special{fp}% \special{pa 2222 2016}% \special{pa 2052 1846}% \special{fp}% \special{pa 2190 2016}% \special{pa 2036 1862}% \special{fp}% \special{pa 2156 2016}% \special{pa 2018 1880}% \special{fp}% \special{pa 2124 2016}% \special{pa 2002 1896}% \special{fp}% \special{pa 2090 2016}% \special{pa 1986 1912}% \special{fp}% \special{pa 2058 2016}% \special{pa 1970 1928}% \special{fp}% \special{pa 2024 2016}% \special{pa 1952 1946}% \special{fp}% \special{pa 1992 2016}% \special{pa 1936 1962}% \special{fp}% \special{pa 1958 2016}% \special{pa 1920 1978}% \special{fp}% \end{picture}% %\input{tsuda2003ics2_zu_2} \end{center} \end{document}