{"id":1417,"date":"2015-04-12T10:28:11","date_gmt":"2015-04-12T09:28:11","guid":{"rendered":"http:\/\/www.unimath.fr\/?p=1417"},"modified":"2016-05-28T19:47:33","modified_gmt":"2016-05-28T18:47:33","slug":"dm-guide-nombres-complexes-3","status":"publish","type":"post","link":"http:\/\/www.unimath.fr\/?p=1417","title":{"rendered":"DM guid\u00e9 Nombres complexes 3"},"content":{"rendered":"

Sujet :\u00a0Cliquer ici<\/a><\/strong><\/p>\n

    \n
  1. Forme exponentielle
    \n[peekaboo_link name=\u00a0\u00bbquestion1″]R\u00e9ponse[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion1″]\\frac{\\sqrt{3}}{2} e^{i \\frac{\\pi}{6}}[\/peekaboo_content]<\/li>\n
  2. a. Preuve suite\u00a0g\u00e9om\u00e9trique
    \n[peekaboo_link name=\u00a0\u00bbquestion3″]Aide[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion3″] r_{n+1} est le module de z_{n+1} puis utiliser la propri\u00e9t\u00e9 des modules qui consiste \u00e0 s\u00e9parer en deux modules quand il y a multiplication[\/peekaboo_content]
    \nb. Expression en fonction de n
    \n[peekaboo_link name=\u00a0\u00bbquestion3b\u00a0\u00bb]Aide [\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion3b\u00a0\u00bb]Formule classique du cours sur les suites g\u00e9om\u00e9triques [\/peekaboo_content]
    \n<\/span>[peekaboo_link name=\u00a0\u00bbquestion4″]R\u00e9ponse<\/span>[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion4″]r_n = \\left(\\frac{\\sqrt{3}}{2}\\right)^n [\/peekaboo_content]
    \nc. Distance OA_n
    \n<\/span>[peekaboo_link name=\u00a0\u00bbquestion4b\u00a0\u00bb]Aide[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion4b\u00a0\u00bb]\u00a0<\/span>\u00a0Traduire cette distance en terme de module[\/peekaboo_content]
    \n<\/span>[peekaboo_link name=\u00a0\u00bbquestion4c\u00a0\u00bb]Aide 2 [\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion4c\u00a0\u00bb]Revoir le cours pour limite de q^n [\/peekaboo_content]
    \n[peekaboo_link name=\u00a0\u00bbquestion5″]R\u00e9ponse [\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion5″]La limite \u00e9tant \u00e9gale \u00e0 0, que peut-on en d\u00e9duire pour le point A_n quand n tend vers +\\infty ?[\/peekaboo_content]<\/span><\/li>\n
  3. Algorithme
    \na. [peekaboo_link name=\u00a0\u00bbquestion10″]Valeur affich\u00e9e\u00a0[\/peekaboo_link]
    \n[peekaboo_content name=\u00a0\u00bbquestion10″]n = 5[\/peekaboo_content]
    \n<\/span>b. R\u00f4le de l’algorithme ?
    \n[peekaboo_link name=\u00a0\u00bbquestion6″]R\u00e9ponse[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion6″]Il affiche le plus petit entier n tel que\u00a0r_n \\leqslant P
    \n[\/peekaboo_content]<\/li>\n
  4. a. Montrer que le triangle est rectangle
    \n[peekaboo_link name=\u00a0\u00bbquestion12″]M\u00e9thode 1 [\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion12″]On peut montrer que l’angle (\\overrightarrow{A_{n+1}O}, \\overrightarrow{A_{n+1}A_{n}}) \u00a0est \u00e9gal \u00e0\u00a0\\frac{\\pi}{2} ou\u00a0\\frac{-\\pi}{2} en utilisant les arguments[\/peekaboo_content]
    \n<\/span>[peekaboo_link name=\u00a0\u00bbquestion13″]M\u00e9thode 2[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion13″] On peut calculer les distances gr\u00e2ce aux modules et utiliser la r\u00e9ciproque du th\u00e9or\u00e8me de Pythagore\u00a0[\/peekaboo_content]
    \nb.\u00a0Valeurs de n pour lesquelles A_n est un point de l’axe des ordonn\u00e9es
    \n[peekaboo_link name=\u00a0\u00bbquestion15″]Aide[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion15″]Le point appartient \u00e0 l’axe des ordonn\u00e9es si et seulement si z_n a pour argument \\frac{\\pi}{2} ou \\frac{-\\pi}{2} modulo 2\\pi ce qui peut se traduire par argument \u00e9gal \u00e0\u00a0\\frac{\\pi}{2} modulo \\pi c’est-\u00e0-dire \\frac{\\pi}{2} + k \\pi avec k \\in Z[\/peekaboo_content]
    \n<\/span>[peekaboo_link name=\u00a0\u00bbquestion16″]R\u00e9ponse[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion16″]n = 3 + 6k avec k entier naturel car n positif [\/peekaboo_content]
    \nc. Construction.
    \n[peekaboo_link name=\u00a0\u00bbquestion17″]Aide 1[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion17″]Remarquer que A_6 appartient \u00e0 l’axe des abscisses et pour les autres points se rappeler que si ABC est rectangle en C alors C appartient au cercle de diam\u00e8tre [BC]. [\/peekaboo_content]
    \n[peekaboo_link name=\u00a0\u00bbquestion18″]Aide 2[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion18″]Utiliser les angles d\u00e9j\u00e0 construits sur la figure et penser \u00e0 prolonger des droites[\/peekaboo_content]<\/li>\n<\/ol>\n

    Correction : cliquer ici<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

    Sujet :\u00a0Cliquer ici Forme exponentielle [peekaboo_link name=\u00a0\u00bbquestion1″]R\u00e9ponse[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion1″][\/peekaboo_content] a. Preuve suite\u00a0g\u00e9om\u00e9trique [peekaboo_link name=\u00a0\u00bbquestion3″]Aide[\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion3″] est le module de puis utiliser la propri\u00e9t\u00e9 des modules qui consiste \u00e0 s\u00e9parer en deux modules quand il y a multiplication[\/peekaboo_content] b. Expression en fonction de n [peekaboo_link name=\u00a0\u00bbquestion3b\u00a0\u00bb]Aide [\/peekaboo_link][peekaboo_content name=\u00a0\u00bbquestion3b\u00a0\u00bb]Formule classique du cours sur les suites g\u00e9om\u00e9triques [\/peekaboo_content] [peekaboo_link […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/www.unimath.fr\/index.php?rest_route=\/wp\/v2\/posts\/1417"}],"collection":[{"href":"http:\/\/www.unimath.fr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.unimath.fr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.unimath.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.unimath.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1417"}],"version-history":[{"count":13,"href":"http:\/\/www.unimath.fr\/index.php?rest_route=\/wp\/v2\/posts\/1417\/revisions"}],"predecessor-version":[{"id":2333,"href":"http:\/\/www.unimath.fr\/index.php?rest_route=\/wp\/v2\/posts\/1417\/revisions\/2333"}],"wp:attachment":[{"href":"http:\/\/www.unimath.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1417"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.unimath.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1417"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.unimath.fr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1417"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}