{"id":1905,"date":"2020-01-19T10:17:14","date_gmt":"2020-01-19T09:17:14","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=1905"},"modified":"2023-10-15T15:43:41","modified_gmt":"2023-10-15T13:43:41","slug":"mesure-principale-dun-angle-avec-python","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/01\/19\/mesure-principale-dun-angle-avec-python\/","title":{"rendered":"Mesure principale d&rsquo;un angle avec Python"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Dans les programmes de 1\u00e8re Sp\u00e9cialit\u00e9 Math, il est question de trigonom\u00e9trie&#8230; Et c&rsquo;est la p\u00e9riode de l&rsquo;ann\u00e9e o\u00f9 les professeur.e.s peuvent aborder cette notion avec leurs \u00e9l\u00e8ves. J&rsquo;ai donc voulu \u00e9crire un programme Python permettant de trouver la mesure principale d&rsquo;un angle s&rsquo;\u00e9crivant sous la forme \\(\\displaystyle\\frac{a\\pi}{b}\\), c&rsquo;est-\u00e0-dire la mesure comprise dans l&rsquo;intervalle \\(]-\\pi;\\pi]\\) \u00e9quivalente \u00e0 celle donn\u00e9e modulo \\(2\\pi\\).<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 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\/19\/mesure-principale-dun-angle-avec-python\/#Rappels_mathematiques\" >Rappels math\u00e9matiques<\/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\/2020\/01\/19\/mesure-principale-dun-angle-avec-python\/#Sur_un_exemple\" >Sur un exemple<\/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\/2020\/01\/19\/mesure-principale-dun-angle-avec-python\/#Cas_general\" >Cas g\u00e9n\u00e9ral<\/a><\/li><\/ul><\/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\/19\/mesure-principale-dun-angle-avec-python\/#En_Python\" >En Python<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"rappels-mathematiques\"><span class=\"ez-toc-section\" id=\"Rappels_mathematiques\"><\/span>Rappels math\u00e9matiques<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"sur-un-exemple\"><span class=\"ez-toc-section\" id=\"Sur_un_exemple\"><\/span>Sur un exemple<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">On consid\u00e8re l&rsquo;angle de mesure \\(\\displaystyle\\frac{101\\pi}{7}\\).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u00c9tape 1 : effectuer la division euclidienne de 101 par 7. Petit flash-back de 6\u00e8me&#8230; on \u00e9crit :  \\(101 = 7 \\times 14 + 3\\). Certains calculatrices permettent d&rsquo;obtenir le quotient (14) et le reste (3), mais sinon, il suffit de prendre la partie enti\u00e8re de \\(101 \\div 7\\), soit de 14,428&#8230; et ensuite de faire \\(101-14\\times7\\) pour obtenir le reste.<\/li>\n\n\n\n<li>\u00c9tape 2 : on revient \u00e0 la mesure de l&rsquo;angle. $$\\frac{101\\pi}{7}=\\frac{(7\\times14+3)\\pi}{7}=14\\pi + \\frac{3\\pi}{7}.$$On remplace le coefficient de \\(\\pi\\) par son \u00e9criture en division euclidienne, puis on coupe la fraction en deux, l&rsquo;int\u00e9r\u00eat \u00e9tant que la premi\u00e8re fraction obtenue est simplifiable et donne le quotient pr\u00e9c\u00e9demment obtenu (ici, 14).<\/li>\n\n\n\n<li>\u00c9tape 3 : on simplifie. Modulo \\(2\\pi\\), \\(14\\pi=0\\) car 14 est un nombre pair (donc, sur le cercle trigonom\u00e9trique, il se place au m\u00eame endroit que 0). Ainsi, l&rsquo;angle trouv\u00e9 est \u00e9gal, modulo \\(2\\pi\\), \u00e0 \\(\\displaystyle\\frac{3\\pi}{7}\\). On vient d&rsquo;obtenir la mesure principale de l&rsquo;angle initial.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Si le quotient de la division euclidienne avait \u00e9t\u00e9 impair, par exemple \\(21\\pi\\), on l&rsquo;aurait remplac\u00e9 par \\(-\\pi\\), puis on l&rsquo;aurait ajout\u00e9 au reste. Par exemple,$$21\\pi+\\frac{2\\pi}{7}=-\\pi+\\frac{2\\pi}{7}=-\\frac{5\\pi}{7}$$Avec cette m\u00e9thode, on obtient toujours la mesure principale de l&rsquo;angle.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"cas-general\"><span class=\"ez-toc-section\" id=\"Cas_general\"><\/span>Cas g\u00e9n\u00e9ral<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">On a un angle de mesure \\(\\displaystyle\\frac{a\\pi}{b}\\). <\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u00c9tape 1 : \\(a = bq + r,\\ 0 \\leq r &lt; b\\)<\/li>\n\n\n\n<li>\u00c9tape 2 : \\(\\displaystyle\\frac{a\\pi}{b}=q\\pi+\\frac{r\\pi}{b}\\)<\/li>\n\n\n\n<li>\u00c9tape 3 : si \\(q\\) est pair, alors la mesure principale est \\(\\displaystyle\\frac{r\\pi}{b}\\), sinon c&rsquo;est \\(-\\pi +  \\displaystyle\\frac{r\\pi}{b} = \\frac{(r-b)\\pi}{b}\\).<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"en-python\"><span class=\"ez-toc-section\" id=\"En_Python\"><\/span>En Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Nous avons maintenant tous les outils pour \u00e9crire un programme en Python. Bien s\u00fbr, il n&rsquo;existe pas qu&rsquo;une seule fa\u00e7on d&rsquo;\u00e9crire le programme. Selon la logique employ\u00e9e, il pourra \u00eatre plus ou moins long.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pour ma part, j&rsquo;ai choisi d&rsquo;ins\u00e9rer une fonction pour placer le point, \u00e0 l&rsquo;aide du module turtle.<\/p>\n\n\n<div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"625\" height=\"383\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/Mesure_principale_Python.png\" alt=\"\" class=\"wp-image-1907\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/Mesure_principale_Python.png 625w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/Mesure_principale_Python-300x184.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/Mesure_principale_Python-600x368.png 600w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">J&rsquo;ai choisi de cr\u00e9er trois fonctions : l&rsquo;une calcule la mesure principale, l&rsquo;autre permet de l&rsquo;afficher (en fonction des coefficients trouv\u00e9s, l&rsquo;affichage doit changer &#8211; cas o\u00f9 le num\u00e9rateur trouv\u00e9 est un multiple du d\u00e9nominateur &#8211;&gt; simplification, etc.) et une derni\u00e8re pour tracer le cercle trigonom\u00e9trique et le point.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"466\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/01\/mesure-principale-python-1024x466.png\" alt=\"mesure principale python\" class=\"wp-image-7254\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/01\/mesure-principale-python-1024x466.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/01\/mesure-principale-python-300x137.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/01\/mesure-principale-python-600x273.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/01\/mesure-principale-python-768x350.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2022\/01\/mesure-principale-python.png 1171w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Le script Python se trouve dans un fichier ZIP ci-dessous:<\/p>\n\n\n\n<div class=\"wp-block-file aligncenter um_article\"><a id=\"wp-block-file--media-684b6e1a-92ea-4fcf-872e-385714e5a050\" href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/mesure-principale.zip\">mesure-principale<\/a><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/01\/mesure-principale.zip\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-684b6e1a-92ea-4fcf-872e-385714e5a050\">T\u00e9l\u00e9charger<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Dans les programmes de 1\u00e8re Sp\u00e9cialit\u00e9 Math, il est question de trigonom\u00e9trie&#8230; Et c&rsquo;est la p\u00e9riode de l&rsquo;ann\u00e9e o\u00f9 les professeur.e.s peuvent aborder cette notion avec leurs \u00e9l\u00e8ves. J&rsquo;ai donc voulu \u00e9crire un programme Python permettant de trouver la mesure principale d&rsquo;un angle s&rsquo;\u00e9crivant sous la forme \\(\\displaystyle\\frac{a\\pi}{b}\\), c&rsquo;est-\u00e0-dire la [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6,5],"tags":[155,150],"class_list":["post-1905","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","category-python","tag-mesure-principale","tag-trigonometrie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Mesure principale d&#039;un angle avec Python - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Comment obtenir la mesure principale d&#039;un angle (en radian) \u00e0 l&#039;aide de Python ? 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