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大阪大学 後期理系 1989年度 問5
解答の著作権は解答作成者に帰属します。無断転載、複製を禁止します。
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 |    |
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\setlength{\textwidth}{420pt}
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\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}
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(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．
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\special{pa 1986 1560}%
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\special{pa 2000 1560}%
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\special{pa 2010 1560}%
\special{pa 2016 1560}%
\special{pa 2020 1560}%
\special{pa 2026 1560}%
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% 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}%
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\special{pa 2036 1830}%
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\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}
Copyright(c) 2021 Infonic Co., Ltd. All rights reserved.