{"id":3010,"date":"2022-05-27T17:14:47","date_gmt":"2022-05-27T15:14:47","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/kosinova-veta-reseny-priklad\/"},"modified":"2022-05-29T19:25:38","modified_gmt":"2022-05-29T17:25:38","slug":"kosinova-veta-reseny-priklad","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/kosinova-veta-reseny-priklad\/","title":{"rendered":"Kosinov\u00e1 v\u011bta \u0159e\u0161en\u00fd p\u0159\u00edklad"},"content":{"rendered":"<p><strong>Zad\u00e1n\u00ed:<\/strong> \u0160esti\u00faheln\u00edk ABCDEF se skl\u00e1d\u00e1 ze dvou \u010dtverc\u016f AXEF a XBCD, rovnostrann\u00e9ho troj\u00faheln\u00edku XDE a tupo\u00fahl\u00e9ho troj\u00faheln\u00edku ABX. D\u00e9lka strany AF je 6 cm. Vypo\u010d\u00edtejte d\u00e9lku strany AB.<\/p>\n<p><strong>\u0158e\u0161en\u00ed:<\/strong><\/p>\n<p>Poj\u010fme si vypsat co zn\u00e1me:<\/p>\n<ul>\n<li>V\u00edme, \u017ee \u0161esti\u00faheln\u00edk<em><strong> ADBCDEF<\/strong><\/em> je tvo\u0159en <strong>dv\u011bma \u010dtverci a dv\u011bma troj\u00faheln\u00edky<\/strong>.<\/li>\n<li>Strany \u010dtverc\u016f <em><strong>AXEF<\/strong><\/em> a <em><strong>XBCD<\/strong><\/em> m\u011b\u0159\u00ed <strong>6 cm<\/strong>. (Pojmenujme si tuto stranu jako stranu <em><strong>a<\/strong><\/em>.)<\/li>\n<li><strong>Rovnostrann\u00fd troj\u00faheln\u00edk <em>XDE<\/em><\/strong> m\u00e1 d\u00e9lku strany rovnou <strong>6 cm<\/strong> a jeho <strong>\u00fahly jsou v\u0161echny stejn\u011b velk\u00e9<\/strong>, tud\u00ed\u017e ka\u017ed\u00fd m\u00e1 <strong>60\u00b0<\/strong>.<\/li>\n<\/ul>\n<p>Pokud bychom v tupo\u00fahl\u00e9m troj\u00faheln\u00edku <em><strong>ABX<\/strong><\/em> vypo\u010d\u00edtali velikost \u00fahlu, kter\u00fd le\u017e\u00ed u vrcholu <em><strong>X (\u00fahel AXB)<\/strong><\/em>, mohli bychom s pomoc\u00ed kosinov\u00e9 v\u011bty dopo\u010d\u00edtat chyb\u011bj\u00edc\u00ed stranu <strong><em>x<\/em><\/strong> (<em><strong>AB<\/strong><\/em>).<\/p>\n<p>A proto\u017ee zn\u00e1me velikosti \u00fahl\u016f <strong><em>AXE<\/em><\/strong>,<em><strong>EXD<\/strong><\/em> a <em><strong>DXB<\/strong><\/em>, m\u016f\u017eeme si velikost \u00fahlu <em><strong>AXB<\/strong><\/em> (kter\u00fd si ozna\u010d\u00edme p\u00edsmenem lambda<strong><em> \u039b<\/em><\/strong>) snadno dopo\u010d\u00edtat. Slo\u017een\u00edm v\u0161ech 4 \u00fahl\u016f toti\u017e z\u00edsk\u00e1me <strong>pln\u00fd \u00fahel<\/strong>, kter\u00fd se rovn\u00e1 <strong>360\u00b0<\/strong>. Sta\u010d\u00ed tedy vypo\u010d\u00edtat n\u00e1sleduj\u00edc\u00ed rovnost:<\/p>\n<p style=\"text-align: center\"><em><strong>\u039b = velikost pln\u00e9ho \u00fahlu-velikost \u00fahlu AXE-velikost \u00fahlu EXD &#8211; velikost \u00fahlu DXB<\/strong><\/em><\/p>\n<p style=\"text-align: center\"><em><strong>\u039b\u00a0= 360\u00b0- 90\u00b0- 60\u00b0- 90\u00b0<\/strong><\/em><\/p>\n<p style=\"text-align: center\"><em><strong>\u039b= 120\u00b0<\/strong><\/em><\/p>\n<p>Nyn\u00ed ji\u017e zn\u00e1me dv\u011b strany a \u00fahel, kter\u00fd spolu sv\u00edraj\u00ed, tud\u00ed\u017e m\u016f\u017eeme vyu\u017e\u00edt kosinovou v\u011btu, kter\u00e1 zn\u00ed:<\/p>\n<p>$$c^2 = a^2 + b^2 &#8211; 2 a b \\cdot \\cos \\gamma$$<\/p>\n<p style=\"text-align: center\">Pokud si vzorec uprav\u00edme na n\u00e1\u0161 p\u0159\u00edklad, dostaneme:<\/p>\n<p>$$x^2 = a^2 + a^2 &#8211; 2 a \\cdot a \\cdot \\cos \\lambda$$<\/p>\n<p style=\"text-align: center\">Nejprve si vyj\u00e1d\u0159\u00edme <strong><em>x<\/em><\/strong> a n\u00e1sledn\u011b do vzorce dosad\u00edme a vypo\u010d\u00edt\u00e1me<\/p>\n<p style=\"text-align: center\">$$x =\\sqrt{ a^2 + a^2 &#8211; 2 a \\cdot a \\cdot \\cos \\lambda}$$<\/p>\n<p>$$x =\\sqrt{ 2a^2 -2 a^2 \\cdot \\cos \\lambda}$$<\/p>\n<p>$$x =\\sqrt{ 2\\cdot 6^2 -2\\cdot 6^2 \\cdot \\cos120\u00b0}$$<\/p>\n<p>$$x=\\sqrt108$$<\/p>\n<p>$$x=\\sqrt{2\\cdot54}=\\sqrt{2\\cdot2\\cdot27}=\\sqrt{2^2\\cdot3^3}=6\\sqrt3$$<\/p>\n<p style=\"text-align: center\">D\u00e9lka strany AB se rovn\u00e1 6\u221a3.<\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-3010","lesson","type-lesson","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Kosinov\u00e1 v\u011bta \u0159e\u0161en\u00fd p\u0159\u00edklad | Studijni-svet.cz<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/kosinova-veta-reseny-priklad\/\" \/>\n<meta property=\"og:locale\" content=\"cs_CZ\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Kosinov\u00e1 v\u011bta \u0159e\u0161en\u00fd p\u0159\u00edklad | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"Zad\u00e1n\u00ed: \u0160esti\u00faheln\u00edk ABCDEF se skl\u00e1d\u00e1 ze dvou \u010dtverc\u016f AXEF a XBCD, rovnostrann\u00e9ho troj\u00faheln\u00edku XDE a tupo\u00fahl\u00e9ho troj\u00faheln\u00edku ABX. D\u00e9lka strany AF je 6 cm. Vypo\u010d\u00edtejte d\u00e9lku strany AB. \u0158e\u0161en\u00ed: Poj\u010fme si vypsat co zn\u00e1me: V\u00edme, \u017ee \u0161esti\u00faheln\u00edk ADBCDEF je tvo\u0159en dv\u011bma \u010dtverci a dv\u011bma troj\u00faheln\u00edky. Strany \u010dtverc\u016f AXEF a XBCD m\u011b\u0159\u00ed 6 cm. (Pojmenujme si ... 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D\u00e9lka strany AF je 6 cm. Vypo\u010d\u00edtejte d\u00e9lku strany AB. \u0158e\u0161en\u00ed: Poj\u010fme si vypsat co zn\u00e1me: V\u00edme, \u017ee \u0161esti\u00faheln\u00edk ADBCDEF je tvo\u0159en dv\u011bma \u010dtverci a dv\u011bma troj\u00faheln\u00edky. Strany \u010dtverc\u016f AXEF a XBCD m\u011b\u0159\u00ed 6 cm. (Pojmenujme si ... 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