{"id":9580,"date":"2024-04-20T17:22:33","date_gmt":"2024-04-20T15:22:33","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=9580"},"modified":"2024-04-20T17:22:35","modified_gmt":"2024-04-20T15:22:35","slug":"trouver-la-mesure-dun-angle-dans-un-triangle","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/04\/20\/trouver-la-mesure-dun-angle-dans-un-triangle\/","title":{"rendered":"Trouver la mesure d&#8217;un angle dans un triangle"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph\">Cet exercice, qui consiste \u00e0 trouver la mesure d&#8217;un angle dans un triangle o\u00f9 l&#8217;on connait la mesure de deux c\u00f4t\u00e9s et celle d&#8217;un autre angle, a \u00e9t\u00e9 propos\u00e9 lors d&#8217;un test \u00e0 des \u00e9l\u00e8ve suisses, dont le niveau \u00e9quivaut \u00e0 celui de 1\u00e8re en France.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure data-wp-context=\"{&quot;imageId&quot;:&quot;69d10e7e8a07d&quot;}\" data-wp-interactive=\"core\/image\" data-wp-key=\"69d10e7e8a07d\" class=\"aligncenter size-medium wp-lightbox-container\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"129\" data-wp-class--hide=\"state.isContentHidden\" data-wp-class--show=\"state.isContentVisible\" data-wp-init=\"callbacks.setButtonStyles\" data-wp-on--click=\"actions.showLightbox\" data-wp-on--load=\"callbacks.setButtonStyles\" data-wp-on-window--resize=\"callbacks.setButtonStyles\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/04\/triangle-300x129.png\" alt=\"trouver la mesure d'un angle dans un triangle\" class=\"wp-image-9581\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/04\/triangle-300x129.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/04\/triangle-600x257.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/04\/triangle.png 690w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><button\n\t\t\tclass=\"lightbox-trigger\"\n\t\t\ttype=\"button\"\n\t\t\taria-haspopup=\"dialog\"\n\t\t\taria-label=\"Agrandir\"\n\t\t\tdata-wp-init=\"callbacks.initTriggerButton\"\n\t\t\tdata-wp-on--click=\"actions.showLightbox\"\n\t\t\tdata-wp-style--right=\"state.imageButtonRight\"\n\t\t\tdata-wp-style--top=\"state.imageButtonTop\"\n\t\t>\n\t\t\t<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12\" height=\"12\" fill=\"none\" viewBox=\"0 0 12 12\">\n\t\t\t\t<path fill=\"#fff\" d=\"M2 0a2 2 0 0 0-2 2v2h1.5V2a.5.5 0 0 1 .5-.5h2V0H2Zm2 10.5H2a.5.5 0 0 1-.5-.5V8H0v2a2 2 0 0 0 2 2h2v-1.5ZM8 12v-1.5h2a.5.5 0 0 0 .5-.5V8H12v2a2 2 0 0 1-2 2H8Zm2-12a2 2 0 0 1 2 2v2h-1.5V2a.5.5 0 0 0-.5-.5H8V0h2Z\" \/>\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n<\/div>\n\n\n<!--more-->\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">Je dois bien dire que cela m&#8217;a laiss\u00e9 perplexe au premier abord car ce n&#8217;\u00e9tait que la premi\u00e8re question sur deux.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">N&#8217;\u00e9tant pas habitu\u00e9 au niveau d&#8217;exigence de l&#8217;enseignement suisse, je ne savais pas trop si la solution devait \u00eatre &#8220;\u00e9vidente&#8221; ou pas&#8230; Quoi qu&#8217;il en soit, pour moi, elle n&#8217;\u00e9tait pas imm\u00e9diate!<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/04\/20\/trouver-la-mesure-dun-angle-dans-un-triangle\/#Comment_trouver_la_mesure_de_langle_manquant_dans_ce_triangle\" >Comment trouver la mesure de l&#8217;angle manquant dans ce triangle?<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/04\/20\/trouver-la-mesure-dun-angle-dans-un-triangle\/#Trouver_la_mesure_de_langle_dans_le_triangle_on_commence_par_calculer_AC\" >Trouver la mesure de l&#8217;angle dans le triangle: on commence par calculer AC<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/04\/20\/trouver-la-mesure-dun-angle-dans-un-triangle\/#Trouver_la_mesure_de_langle_demande_dans_le_triangle_ABC\" >Trouver la mesure de l&#8217;angle demand\u00e9 dans le triangle ABC<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/04\/20\/trouver-la-mesure-dun-angle-dans-un-triangle\/#Reflexions\" >R\u00e9flexions<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Comment_trouver_la_mesure_de_langle_manquant_dans_ce_triangle\"><\/span>Comment trouver la mesure de l&#8217;angle manquant dans ce triangle?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">La configuration propos\u00e9e m&#8217;a fait pens\u00e9 imm\u00e9diatement au th\u00e9or\u00e8me d&#8217;Al-Kashi. Mais il me fallait en premier lieu calculer la mesure du c\u00f4t\u00e9 AC&#8230; et pour cela, ce m\u00eame th\u00e9or\u00e8me \u00e9tait aussi utilisable.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trouver_la_mesure_de_langle_dans_le_triangle_on_commence_par_calculer_AC\"><\/span>Trouver la mesure de l&#8217;angle dans le triangle: on commence par calculer AC<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">D&#8217;apr\u00e8s le th\u00e9or\u00e8me d&#8217;Al-Kashi, on peut \u00e9crire:$$\\begin{align}AB^2 &amp; =  CA^2+CB^2-2\\times CA \\times CB \\times \\cos(15^\\circ)\\\\\\iff 4,5^2 &amp; = AC^2 + 6^2 &#8211; 12AC\\cos(15^\\circ)\\\\\\iff &amp; AC^2 &#8211; 12\\cos(15^\\circ)AC + 15,75=0\\end{align}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">\\(AC\\) est alors une des solutions de cette derni\u00e8re \u00e9quation du second degr\u00e9, de discriminant:$$\\begin{align}\\Delta &amp; =144\\cos^2(15^\\circ)-63\\\\ &amp; = 144\\times\\left(\\frac{\\sqrt2+\\sqrt6}{4}\\right)^2-63\\\\&amp;=36\\sqrt3+9.\\end{align}$$On trouve alors deux solutions \u00e0 l&#8217;\u00e9quation:$$\\begin{align} x_1 &amp; = \\frac{12\\cos(15^\\circ)-\\sqrt{ 36\\sqrt3+9 }}{2} \\\\  &amp; = \\frac{3\\big(\\sqrt2+\\sqrt6\\big)-\\sqrt{36\\sqrt3+9}}{2} \\\\x_1 &amp; \\approx 1,57\\end{align}$$et$$\\begin{align}x_2 &amp; = \\frac{3\\big(\\sqrt2+\\sqrt6\\big)+\\sqrt{36\\sqrt3+9}}{2}\\\\ &amp; \\approx10,02.\\end{align}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">Il va de soit que seule la derni\u00e8re valeur convient \u00e0 notre probl\u00e8me. On a alors:$$AC \\approx 10,02.$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Trouver_la_mesure_de_langle_demande_dans_le_triangle_ABC\"><\/span>Trouver la mesure de l&#8217;angle demand\u00e9 dans le triangle ABC<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">Maintenant que nous avons \\(AC\\), utilisons \u00e0 nouveau le th\u00e9or\u00e8me d&#8217;Al-Kashi pour trouver l&#8217;angle demand\u00e9:$$\\begin{align}BC^2 &amp; = AB^2 + AC^2 &#8211; 2\\times AB \\times AC \\times\\cos(\\widehat{A}) \\\\ \\iff 36 &amp; \\approx 4,5^2 + 10,02^2 &#8211; 2\\times4,5\\times10,02\\times\\cos(\\widehat{A}) \\\\ \\iff \\cos(\\widehat{A}) &amp; \\approx 0,938682634731\\\\\\iff \\widehat{A} &amp; \\approx 20^\\circ.\\end{align}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Reflexions\"><\/span>R\u00e9flexions<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">Vous vous demandez peut-\u00eatre pourquoi je m&#8217;emb\u00eate \u00e0 garder autant de valeurs exactes et autant de d\u00e9cimales jusqu&#8217;au r\u00e9sultat final, non ?<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph\">Si tel est le cas, je vous rassure: ce n&#8217;est pas parce que je suis fou&#8230; \ud83d\ude42 mais parce que d\u00e8s que nous donnons une valeur approch\u00e9e &#8220;trop arrondie&#8221;, le r\u00e9sultat final n&#8217;est pas celui attendu. Croyez-moi sur parole: j&#8217;ai essay\u00e9 d&#8217;arrondir \u00e0 plusieurs endroits, et je n&#8217;arrivais jamais exactement \u00e0 20 degr\u00e9s, mais \u00e0 17 degr\u00e9s par exemple. C&#8217;est dingue comme notre fa\u00e7on d&#8217;arrondir peut avoir une incidence majeure sur les calculs de mesures d&#8217;angles!<\/p>\n\n\n\n<p>Si vous avez une solution plus simple, n&#8217;h\u00e9sitez pas \u00e0 la proposer en commentaire! Car \u00e0 mon avis, il y a en effet plus simple&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Cet exercice, qui consiste \u00e0 trouver la mesure d&#8217;un angle dans un triangle o\u00f9 l&#8217;on connait la mesure de deux c\u00f4t\u00e9s et celle d&#8217;un autre angle, a \u00e9t\u00e9 propos\u00e9 lors d&#8217;un test \u00e0 des \u00e9l\u00e8ve suisses, dont le niveau \u00e9quivaut \u00e0 celui de 1\u00e8re en France.<\/p>\n","protected":false},"author":1,"featured_media":9596,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[49,340,46],"class_list":["post-9580","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-al-kashi","tag-angle","tag-triangle"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Trouver la mesure d&#039;un angle dans un triangle - 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