{"id":3316,"date":"2025-09-28T00:25:57","date_gmt":"2025-09-27T15:25:57","guid":{"rendered":"https:\/\/math-friend.com\/?p=3316"},"modified":"2025-09-28T00:26:00","modified_gmt":"2025-09-27T15:26:00","slug":"%e3%80%90%e6%9d%b1%e4%ba%ac%e5%a4%a7%e5%ad%a6-%e6%96%87%e7%b3%bb1-2010%e3%80%91%e4%b8%89%e8%a7%92%e9%96%a2%e6%95%b0%e3%81%a8%e4%b8%89%e8%a7%92%e5%bd%a2%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%81%ae%e6%9c%80","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3316","title":{"rendered":"\u3010\u6771\u4eac\u5927\u5b66 \u6587\u7cfb1 (2010)\u3011\u4e09\u89d2\u95a2\u6570\u3068\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u306e\u6700\u5927\u5316"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(\\mathrm{O}\\)\u3092\u539f\u70b9\u3068\u3059\u308b\u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u70b9\\(\\mathrm{A}(-3,0)\\)\u3092\u3068\u308a, \\(0^\\circ < \\theta < 120^\\circ\\)\u306e\u7bc4\u56f2\u306b\u3042\u308b\\(\\theta\\)\u306b\u5bfe\u3057\u3066, \u6b21\u306e\u6761\u4ef6\\((\\mathrm{i})\\), \\((\\mathrm{ii})\\)\u3092\u307f\u305f\u3059\\(2\\)\u70b9\\(\\mathrm{B}\\), \\(\\mathrm{C}\\)\u3092\u8003\u3048\u308b.
\\((\\mathrm{i})\\) \\(\\mathrm{B}\\)\u306f\\(y>0\\)\u306e\u90e8\u5206\u306b\u3042\u308a, \\(\\mathrm{OB}=2\\)\u304b\u3064\\(\\angle{\\mathrm{AOB}}=180^\\circ-\\theta\\)\u3067\u3042\u308b.
\\((\\mathrm{ii})\\) \\(\\mathrm{C}\\)\u306f\\(y<0\\)\u306e\u90e8\u5206\u306b\u3042\u308a, \\(\\mathrm{OC}=1\\)\u304b\u3064\\(\\angle{\\mathrm{BOC}}=120^\\circ\\)\u3067\u3042\u308b. \u305f\u3060\u3057, \\(\\triangle\\mathrm{ABC}\\)\u306f\\(\\mathrm{O}\\)\u3092\u542b\u3080\u3082\u306e\u3068\u3059\u308b.
\u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(\\triangle\\mathrm{OAB}\\)\u3068\\(\\triangle\\mathrm{OAC}\\)\u306e\u9762\u7a4d\u304c\u7b49\u3057\u3044\u3068\u304d, \\(\\theta\\)\u306e\u5024\u3092\u6c42\u3081\u3088.
(2) \\(\\theta\\)\u3092\\(0^\\circ < \\theta < 120^\\circ\\)\u306e\u7bc4\u56f2\u3067\u52d5\u304b\u3059\u3068\u304d, \\(\\triangle\\mathrm{OAB}\\)\u3068\\(\\triangle\\mathrm{OAC}\\)\u306e\u9762\u7a4d\u306e\u548c\u304c\u6700\u5927\u5024\u3068, \u305d\u306e\u3068\u304d\u306e\\(\\sin{\\theta}\\)\u306e\u5024\u3092\u6c42\u3081\u3088.
(2010 \u6771\u4eac\u5927\u5b66 \u6587\u7cfb [1])<\/span><\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \\(\\mathrm{B}\\), \\(\\mathrm{C}\\)\u3092\u56f3\u793a\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a, \u5ea7\u6a19\u306f\u305d\u308c\u305e\u308c, \\((2\\cos{\\theta}, 2\\sin{\\theta})\\), \\((\\cos{(\\theta-120^\\circ)}, \\sin{(\\theta-120^\\circ)})\\)\u3068\u306a\u308b.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\\(\\triangle\\mathrm{OAB}\\)\u3068\\(\\triangle\\mathrm{OAC}\\)\u306f\u8fba\\(\\mathrm{OA}\\)\u3092\u5171\u6709\u3057, \u305d\u306e\u9762\u7a4d\u304c\u7b49\u3057\u304f\u306a\u308b\u306e\u306f, \u8fba\\(\\mathrm{OA}\\)\u3092\u5e95\u8fba\u3068\u3057\u3066\u898b\u305f\u3068\u304d\u306e\u9ad8\u3055\\(\\mathrm{B}\\), \\(\\mathrm{C}\\)\u306e\\(y\\)\u5ea7\u6a19\u306e\u7d76\u5bfe\u5024\u304c\u7b49\u3057\u3044\u3068\u304d\u3067\u3042\u308b. \\(\\mathrm{B}\\)\u306e\\(y\\)\u5ea7\u6a19\u306f\u5e38\u306b\u6b63, \\(\\mathrm{C}\\)\u306e\\(y\\)\u5ea7\u6a19\u306f\u5e38\u306b\u8ca0\u306a\u306e\u3067, \u3053\u306e\u6761\u4ef6\u306f,$$
2\\sin{\\theta}=-\\sin{(\\theta-120^\\circ)}
$$\u3067\u3042\u308b. \u3053\u308c\u3092\u5909\u5f62\u3059\u308b\u3068,$$
\\begin{align}
2\\sin{\\theta}&=\\frac{1}{2}\\sin{\\theta}+\\frac{\\sqrt{3}}{2}\\cos{\\theta}\\\\[1.5ex]
\\frac{3}{2}\\sin{\\theta}&=\\frac{\\sqrt{3}}{2}\\cos{\\theta}\\\\[1.5ex]
\\sqrt{3}\\sin{\\theta}&=\\cos{\\theta}
\\end{align}
$$\u3067\u3042\u308b. \\(\\theta=90^\\circ\\)\u306f\u3053\u306e\u65b9\u7a0b\u5f0f\u306e\u89e3\u3067\u306f\u306a\u3044\u306e\u3067, \\(\\theta\\neq 90^\\circ\\)\u3067\u3042\u308b. \u3053\u306e\u3068\u304d, \\(\\cos{\\theta}\\neq 0\\)\u3067\u3042\u308b\u304b\u3089, \u4e21\u8fba\u3092\\(\\sqrt{3}\\cos{\\theta}\\)\u3067\u5272\u3063\u3066,$$
\\tan{\\theta}=\\frac{1}{\\sqrt{3}}
$$\u3068\u306a\u308a, \\(0^\\circ < \\theta < 120^\\circ\\)\u3088\u308a, \\(\\theta=30^\\circ\\)\u3067\u3042\u308b.<\/p>\n\n\n\n

(2) \\(\\triangle\\mathrm{OAB}\\)\u3068\\(\\triangle\\mathrm{OAC}\\)\u306e\u9762\u7a4d\u306e\u548c\\(S\\)\u306f,$$
S=\\frac{1}{2}\\times 3\\times2\\sin{\\theta}+\\frac{1}{2}\\times 3\\times \\left\\{-\\sin{(\\theta-120^\\circ)}\\right\\}
$$\u3067\u3042\u308a, \u3053\u308c\u304b\u3089, $$
\\begin{align}
S&=3\\sin{\\theta}+\\frac{3}{2}\\left(\\frac{1}{2}\\sin{\\theta}+\\frac{\\sqrt{3}}{2}\\cos{\\theta}\\right)\\\\[1.5ex]
&=\\frac{15}{4}\\sin{\\theta}+\\frac{3\\sqrt{3}}{4}\\cos{\\theta}\\\\[1.5ex]
&=\\frac{3}{4}\\left(5\\sin{\\theta}+\\sqrt{3}\\cos{\\theta}\\right)\\\\[1.5ex]&=\\frac{3}{4}\\sqrt{5^2+3}\\left(\\sin{\\theta}\\cdot \\frac{5}{2\\sqrt{7}}+\\cos{\\theta}\\cdot\\frac{\\sqrt{3}}{2\\sqrt{7}}\\right)\\\\[1.5ex]
&=\\frac{3\\sqrt{7}}{2}\\left(\\sin{\\theta}\\cos{\\alpha}+\\cos{\\theta}\\sin{\\alpha}\\right)\\\\[1.5ex]
&=\\frac{3\\sqrt{7}}{2}\\sin{(\\theta+\\alpha)}
\\end{align}
$$\u3067\u3068\u306a\u308b. \u3053\u3053\u3067, \\(\\alpha\\)\u306f\\(\\displaystyle \\cos{\\alpha}=\\frac{5}{2\\sqrt{7}}\\), \\(\\displaystyle \\sin{\\alpha}=\\frac{\\sqrt{3}}{2\\sqrt{7}}\\)\u3092\u6e80\u305f\u3059\u89d2\u5ea6\u3067\u3042\u308b.

\\(\\cos{\\alpha}>0\\), \\(\\sin{\\alpha}>0\\)\u3088\u308a, \\(0^\\circ <\\alpha < 90^\\circ\\)\u304c\u308f\u304b\u308a, \u3053\u308c\u304b\u3089,$$
0^\\circ<\\alpha<\\theta+\\alpha<\\alpha+120^\\circ<210^\\circ
$$\u3068\u306a\u308b. \\(0^\\circ<\\theta+\\alpha<210^\\circ\\)\u306e\u7bc4\u56f2\u3067\u306f, \\(S\\)\u306f\\(\\theta+\\alpha=90^\\circ\\)\u306e\u3068\u304d\u306b, \u6700\u5927\u3068\u306a\u308b. \\(0^\\circ<\\theta(=90^\\circ -\\alpha)<90^\\circ\\)\u3088\u308a, \u3053\u308c\u3092\u6e80\u305f\u3059\\(\\theta\\)\u306f\u305f\u3057\u304b\u306b\\(0^\\circ < \\theta < 120^\\circ\\)\u306e\u7bc4\u56f2\u306b\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308b.

\\(S\\)\u306e\u6700\u5927\u5024\u306f\\( \\displaystyle \\frac{3\\sqrt{7}}{2}\\)\u3067\u3042\u308a, \u3053\u306e\u3068\u304d, \\(\\sin{\\theta}\\)\u306f,$$
\\sin{\\theta}=\\sin{(90^\\circ – \\alpha)}=\\cos{\\alpha}=\\frac{5}{2\\sqrt{7}}=\\frac{5\\sqrt{7}}{14}
$$\u3068\u306a\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n