{"id":3067,"date":"2025-09-07T00:16:42","date_gmt":"2025-09-06T15:16:42","guid":{"rendered":"https:\/\/math-friend.com\/?p=3067"},"modified":"2025-09-07T00:18:42","modified_gmt":"2025-09-06T15:18:42","slug":"%e3%80%90%e4%b8%80%e6%a9%8b%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e6%9c%80%e5%a4%a7%e8%a7%92120%e3%81%ae%e4%b8%89%e8%a7%92%e5%bd%a2%e3%81%a7%e4%bd%99%e5%bc%a6%e3%81%ae%e5%80%a4%e3%82%92","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3067","title":{"rendered":"\u3010\u4e00\u6a4b\u5927\u5b66\u5165\u8a66\u3011\u6700\u5927\u89d2120\u00b0\u306e\u4e09\u89d2\u5f62\u3067\u4f59\u5f26\u306e\u5024\u3092\u6c42\u3081\u308b(2014)"},"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

\u5e73\u9762\u4e0a\u306e\\(4\\)\u70b9\\(\\mathrm{O}\\), \\(\\mathrm{A}\\), \\(\\mathrm{B}\\), \\(\\mathrm{C}\\)\u304c, \\(\\mathrm{OA}=4\\), \\(\\mathrm{OB}=3\\), \\(\\mathrm{OC}=2\\), \\(\\overrightarrow{\\mathrm{OB}}\\cdot\\overrightarrow{\\mathrm{OC}}=3\\)\u3092\u6e80\u305f\u3059\u3068\u304d, \\(\\triangle{\\mathrm{ABC}}\\)\u306e\u9762\u7a4d\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.
(2013 \u4e00\u6a4b\u5927\u5b66 [2])<\/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

\\(\\overrightarrow{\\mathrm{OB}}\\cdot\\overrightarrow{\\mathrm{OC}}=3\\)\u3088\u308a,$$
\\begin{align}
|\\overrightarrow{\\mathrm{OB}}||\\overrightarrow{\\mathrm{OC}}|\\cos{\\angle{\\mathrm{BOC}}}&=3\\\\[1.5ex]
3\\cdot 2\\cdot \\cos{\\angle{\\mathrm{BOC}}}&=3\\\\[1.5ex]
\\cos{\\angle{\\mathrm{BOC}}}&=\\frac{1}{2}
\\end{align}
$$\u3068\u306a\u308a, \\(\\angle{\\mathrm{BOC}}=60^\\circ\\)\u304c\u308f\u304b\u308b. \u3053\u308c\u304b\u3089, \\(xy\\)\u5e73\u9762\u4e0a\u3067\\(\\mathrm{B}(3,0)\\), \\(\\mathrm{C}(1,\\sqrt{3})\\)\u3068\u3057\u3066\u3082\u4e00\u822c\u6027\u3092\u5931\u308f\u306a\u3044.

\\(\\mathrm{A}\\)\u306f\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84\\(4\\)\u306e\u5186\u5468\u4e0a\u306b\u3042\u308a, \\(\\overrightarrow{\\mathrm{OA}}\\)\u304c\\(x\\)\u8ef8\u306e\u6b63\u306e\u65b9\u5411\u3068\u306a\u3059\u89d2\u306e\u5927\u304d\u3055\u3092\\(\\theta\\)\u3068\u3059\u308b\u3068, \\(\\mathrm{A}(4\\cos{\\theta}, 4\\sin{\\theta})\\)\u3068\u8868\u305b\u308b. \u3053\u308c\u304b\u3089, $$
\\begin{align}
\\overrightarrow{\\mathrm{BA}}&=(4\\cos{\\theta}-3,4\\sin{\\theta})\\\\[1.5ex]
\\overrightarrow{\\mathrm{BC}}&=(-2, \\sqrt{3})
\\end{align}
$$\u3068\u306a\u308a, \\(\\triangle{\\mathrm{ABC}}\\)\u306e\u9762\u7a4d\\(S\\)\u306f,$$
\\begin{align}
S&=\\frac{1}{2}\\left|(4\\cos{\\theta}-3)\\cdot \\sqrt{3}-4\\sin{\\theta}\\cdot(-2)\\right|\\\\[1.5ex]
&=2\\left|\\sqrt{3}\\cos{\\theta}+2\\sin{\\theta}-\\frac{3\\sqrt{3}}{4}\\right|
\\end{align}
$$\u3068\u8868\u305b\u308b.

\u3053\u3053\u3067, \\(\\sqrt{3}\\cos{\\theta}+2\\sin{\\theta}\\)\u306f\u4e09\u89d2\u95a2\u6570\u306e\u5408\u6210\u304b\u3089,$$
\\begin{align}
\\sqrt{3}\\cos{\\theta}+2\\sin{\\theta}&=\\sqrt{7}\\left(\\sin{\\theta}\\cdot\\frac{2}{\\sqrt{7}}+\\cos{\\theta}\\cdot\\frac{\\sqrt{3}}{\\sqrt{7}}\\right)\\\\[1.5ex]
&=\\sqrt{7}\\sin{(\\theta+\\alpha)}
\\end{align}
$$\u3068\u8868\u305b\u308b. \u3053\u3053\u3067\\(\\alpha\\)\u306f, \\(\\displaystyle \\cos{\\alpha}=\\frac{2}{\\sqrt{7}}\\), \\(\\displaystyle \\sin{\\alpha}=\\frac{\\sqrt{3}}{\\sqrt{7}}\\)\u3092\u6e80\u305f\u3059\u5b9f\u6570\u3067\u3042\u308b.

\u3088\u3063\u3066\\(S\\)\u306f,$$
S=2\\left|\\sqrt{7}\\sin{(\\theta+\\alpha)}-\\frac{3\\sqrt{3}}{4}\\right|
$$\u3068\u306a\u308a, \u3053\u308c\u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f, \\(\\sqrt{7}\\sin{(\\theta+\\alpha)}\\)\u304c\\(-\\sqrt{7}\\)\u3068\u306a\u308b\u3068\u304d\u3067, \u305d\u306e\u6700\u5927\u5024\u306f, \\(\\displaystyle \\frac{4\\sqrt{7}+3\\sqrt{3}}{2}\\)\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