(1)<\/strong>\u3000\\(S \\left( \\dfrac{\\pi}{6} \\right)\\) , \\(V \\left( \\dfrac{\\pi}{6} \\right)\\) \u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) \u306b\u304a\u3051\u308b \\(S( \\theta )\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) \u306b\u304a\u3051\u308b \\(V( \\theta )\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u70b9 A \uff5e D \u305d\u308c\u305e\u308c\u306e, \\(xy\\) \u5e73\u9762\u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u70b9\u3092 A' \uff5e D' \u3068\u304a\u304f\u3068, \u56db\u9762\u4f53 ABCD \u306f, \u76f4\u65b9\u4f53 AC'BD'-A'CB'D \u304b\u3089, \\(4\\) \u3064\u306e\u4e09\u89d2\u3059\u3044 A'ACD , B'BCD , C'CAB , D'DAB \u3092\u9664\u3044\u305f\u7acb\u4f53\u3067\u3042\u308b. \\[\r\nS( \\theta ) = \\dfrac{1}{2} \\cdot 2 \\cos \\theta \\cdot 2 \\sin \\theta = \\sin 2 \\theta \\ .\r\n\\]\r\n\u3088\u3063\u3066, \\(\\theta = \\dfrac{\\pi}{6}\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nV \\left( \\dfrac{\\pi}{6} \\right) & = \\dfrac{8}{3} \\left( \\dfrac{\\sqrt{3}}{2} \\right)^2 \\cdot \\dfrac{1}{2} = \\underline{1} , \\\\\r\nS \\left( \\dfrac{\\pi}{6} \\right) & = \\underline{\\dfrac{\\sqrt{3}}{2}} \\ .\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(0 \\lt 2 \\theta \\lt \\pi\\) \u306a\u306e\u3067, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f \\(\\theta = \\dfrac{\\pi}{4}\\) \u306e\u3068\u304d\r\n\\[\r\n\\underline{1} \\ .\r\n\\]\r\n (3)<\/strong><\/p>\r\n \\[\r\nV( \\theta ) = \\dfrac{8}{3} \\underline{\\left( \\sin \\theta -\\sin^3 \\theta \\right)} _ {[1]} \\ .\r\n\\]\r\n\\(s = \\sin \\theta \\ ( 0 \\lt s \\lt 1 )\\) \u3068\u304a\u3044\u3066, [1] \u3092 \\(f(s) = s -s^3\\) \u3068\u304a\u3051\u3070, \\(f(s)\\) \u304c\u6700\u5927\u306e\u3068\u304d, \\(V( \\theta )\\) \u3082\u6700\u5927\u306b\u306a\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nV( \\theta ) & = ( 2 \\cos \\theta )^2 \\cdot 2 \\sin \\theta -4 \\cdot \\dfrac{1}{3} \\cdot \\dfrac{1}{2} ( 2 \\cos \\theta )^2 \\cdot 2 \\sin \\theta \\\\\r\n& = \\dfrac{1}{3} ( 2 \\cos \\theta )^2 \\cdot 2 \\sin \\theta = \\dfrac{8}{3} \\cos^2 \\theta \\sin \\theta \\ .\r\n\\end{align}\\]\r\n\u307e\u305f, \\(xz\\) \u5e73\u9762\u306b\u3088\u308b\u5207\u308a\u53e3\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u3068\u306a\u308b\u306e\u3067<\/p>\r\n![\"iks20140201\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/iks20140201.svg\")
\r\n
\r\n\\(f(s)\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nf'(s) = 1 -3s^2 = \\left( 1 -\\sqrt{3} s \\right) \\left( 1 +\\sqrt{3} s \\right) \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(f(s)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccc} s & (0) & \\cdots & \\dfrac{1}{\\sqrt{3}} & \\cdots & (1) \\\\ \\hline f'(s) & & + & 0 & - & \\\\ \\hline f(s) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\\(f(s)\\) \u306e\u6700\u5927\u5024\u306b\u3064\u3044\u3066\r\n\\[\r\nf \\left( \\dfrac{1}{\\sqrt{3}} \\right) = \\dfrac{1}{\\sqrt{3}} -\\dfrac{1}{3 \\sqrt{3}} = \\dfrac{2 \\sqrt{3}}{9} \\ .\r\n\\]\r\n\u3088\u3063\u3066, \\(V( \\theta )\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{8}{3} f \\left( \\dfrac{1}{\\sqrt{3}} \\right) = \\underline{\\dfrac{16 \\sqrt{3}}{27}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\lt \\theta \\lt \\dfrac{\\pi}{2}\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570 \\(\\theta\\) \u306b\u5bfe\u3057, \\(xyz\\) \u7a7a\u9593\u5185\u306e \\(4\\) \u70b9 A \\(( \\cos \\theta , \\cos \\thet … \u7d9a\u304d\u3092\u8aad\u3080