{"id":1925,"date":"2020-01-29T18:10:03","date_gmt":"2020-01-29T17:10:03","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1925"},"modified":"2024-06-09T17:00:11","modified_gmt":"2024-06-09T15:00:11","slug":"preparer-son-devoir-sur-les-nombres-complexes-partie-algebre","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/01\/29\/preparer-son-devoir-sur-les-nombres-complexes-partie-algebre\/","title":{"rendered":"Pr\u00e9parer son devoir sur les nombres complexes, partie Alg\u00e8bre"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Voici quatre exercices qui permettrons aux \u00e9l\u00e8ves de Terminale S de pr\u00e9parer leur devoir sur les nombres complexes (premi\u00e8re partie de la le\u00e7on : l&rsquo;ag\u00e8bre). <\/p>\n\n\n\n<!--more-->\n\n\n\n<p class=\"wp-block-paragraph\">\u00c0 travers ces exercices, tout ce qu&rsquo;il y a \u00e0 savoir est r\u00e9sum\u00e9.<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_88 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\/2020\/01\/29\/preparer-son-devoir-sur-les-nombres-complexes-partie-algebre\/#Exercice_1\" >Exercice 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/01\/29\/preparer-son-devoir-sur-les-nombres-complexes-partie-algebre\/#Exercice_2\" >Exercice 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/01\/29\/preparer-son-devoir-sur-les-nombres-complexes-partie-algebre\/#Exercice_3\" >Exercice 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/01\/29\/preparer-son-devoir-sur-les-nombres-complexes-partie-algebre\/#Exercice_4\" >Exercice 4<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/01\/29\/preparer-son-devoir-sur-les-nombres-complexes-partie-algebre\/#Note\" >Note<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exercice_1\"><\/span>Exercice 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re le polyn\u00f4me complexe: $$P(z)=z^3-\\text{i} z^2- \\text{i}z-1- \\text{i}$$<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Montrer que \\(\\alpha=1+ \\text{i}\\) est une racine de <em>P<\/em>.<\/li>\n\n\n\n<li>En d\u00e9duire la valeur des nombres a, b et c tels que: $$P(z)=(z-\\alpha)(az^2+bz+c).$$<\/li>\n\n\n\n<li>R\u00e9soudre dans \\(\\mathbb{C}\\) l&rsquo;\u00e9quation \\(z^2+z+1=0\\), puis en d\u00e9duire toutes les solutions de l&rsquo;\u00e9quation \\(P(z)=0\\). <\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exercice_2\"><\/span>Exercice 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re les deux nombres complexes: $$z_1 = 2\\left[ \\cos\\left(\\frac{5\\pi}{6}\\right) + \\text{i}\\sin\\left(\\frac{5\\pi}{6}\\right) \\right]\\qquad\\text{et}\\qquad z_2=-\\sqrt{2}+ \\text{i} \\sqrt2.$$ On pose alors: $$Z = \\frac{z_1}{z_2}.$$<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Donner la forme alg\u00e9brique de \\(z_1\\).<\/li>\n\n\n\n<li>Donner la forme trigonom\u00e9trique, puis exponentielle de \\(z_2\\).<\/li>\n\n\n\n<li>Donner la forme alg\u00e9brique de \\((z_1)^6\\). En d\u00e9duire que pour tout entier naturel <em>n<\/em>, \\((z_1)^{6n}\\) est un r\u00e9el.<\/li>\n\n\n\n<li>Calculer |<em>Z<\/em>| et d\u00e9terminer un argument de <em>Z<\/em> \u00e0 \\(2\\pi\\) pr\u00e8s.<\/li>\n\n\n\n<li>D\u00e9terminer la forme alg\u00e9brique de <em>Z<\/em>.<\/li>\n\n\n\n<li>En d\u00e9duire la valeur exacte de \\(\\cos\\left(\\frac{\\pi}{12}\\right)\\) et \\(\\sin\\left(\\frac{\\pi}{12}\\right)\\).<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exercice_3\"><\/span>Exercice 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Soit <em>z<\/em> un nombre complexe quelconque. On pose alors:$$z&rsquo; = \\frac{z^2+\\overline{z}^2}{2}.$$<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Montrer que \\(z&rsquo;\\in\\mathbb{R}\\), quel que soit le nombre complexe <em>z<\/em>.<\/li>\n\n\n\n<li>Comment choisir <em>z<\/em> pour que <em>z<\/em>&lsquo; = 0 ?<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Exercice_4\"><\/span>Exercice 4<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">On pose \\(z_1=\\text{e}^{\\text{i}\\theta_1}\\) et \\(z_2=\\text{e}^{\\text{i}\\theta_2}\\).<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Donner la forme alg\u00e9brique de \\(z_1z_2\\).<\/li>\n\n\n\n<li>Donner la forme trigonom\u00e9trique de \\(z_1z_2\\).<\/li>\n\n\n\n<li>En d\u00e9duire une formule permettant de calculer \\(\\cos(\\theta_1+\\theta_2)\\) en fonction de \\(\\cos\\theta_1\\), \\(\\cos\\theta_2\\), \\(\\sin\\theta_1\\) et \\(\\sin\\theta_2\\).<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Note\"><\/span>Note<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Dans la mesure o\u00f9 je n&rsquo;ai pas fait de corrig\u00e9, si vous d\u00e9cidez de vous entra\u00eener sur ces exercices, vous pouvez proposer \u00e0 votre enseignant.e. (ou \u00e0 toutes autre personne comp\u00e9tente) vos r\u00e9ponses.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Le sujet au format \\(\\LaTeX\\) est disponible ci-dessous:<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/Nombres-complexes-Partie-1.zip\">Nombres complexes Partie 1<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/Nombres-complexes-Partie-1.zip\" class=\"wp-block-file__button wp-element-button\" download>T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Voici quatre exercices qui permettrons aux \u00e9l\u00e8ves de Terminale S de pr\u00e9parer leur devoir sur les nombres complexes (premi\u00e8re partie de la le\u00e7on : l&rsquo;ag\u00e8bre).<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[163],"class_list":["post-1925","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","tag-nombres-complexes"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - 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