{"id":2062,"date":"2020-02-26T10:40:57","date_gmt":"2020-02-26T09:40:57","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2062"},"modified":"2020-09-11T10:17:29","modified_gmt":"2020-09-11T08:17:29","slug":"lien-entre-cos-sin-et-exp","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/26\/lien-entre-cos-sin-et-exp\/","title":{"rendered":"cos sin exp : pourquoi \u00e9crit-on cos(x)+isin(x)=exp(ix)?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Nous allons voir ce qui lit cos, sin et exp. Un <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Nombre_complexe\" target=\"_blank\" rel=\"noreferrer noopener\">nombre complexe<\/a> admet trois \u00e9critures : sa forme alg\u00e9brique (<em>z<\/em> = <em>x<\/em> + i<em>y<\/em>), sa forme trigonom\u00e9trique (<em>z<\/em> = <em>r<\/em>[cos(<em>t<\/em>) + isin(<em>t<\/em>)]) et&#8230; sa forme exponentielle (<em>z<\/em> = exp(i<em>t<\/em>)). Jusqu&rsquo;en 2020, les \u00e9l\u00e8ves de terminale de France voyaient cette derni\u00e8re forme comme parachut\u00e9e. Dans cet article, je vous propose de vous expliquer les raisons pour lesquelles on se permet d&rsquo;utiliser une telle notation.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"701\" height=\"143\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/cos-et-sin.jpg\" alt=\"cos sin exp\" class=\"wp-image-3383\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/cos-et-sin.jpg 701w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/cos-et-sin-300x61.jpg 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/09\/cos-et-sin-600x122.jpg 600w\" sizes=\"auto, (max-width: 701px) 100vw, 701px\" \/><\/figure><\/div>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 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\/02\/26\/lien-entre-cos-sin-et-exp\/#Une_approximation_par_des_polynomes_pour_connaitre_le_lien_entre_cos_sin_et_exp\" >Une approximation par des polyn\u00f4mes pour conna\u00eetre le lien entre cos, sin et exp<\/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\/02\/26\/lien-entre-cos-sin-et-exp\/#La_fonction_sinus\" >La fonction sinus<\/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\/02\/26\/lien-entre-cos-sin-et-exp\/#La_fonction_cosinus\" >La fonction cosinus<\/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\/2020\/02\/26\/lien-entre-cos-sin-et-exp\/#La_fonction_exponentielle\" >La fonction exponentielle<\/a><\/li><\/ul><\/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\/02\/26\/lien-entre-cos-sin-et-exp\/#Les_puissances_de_%C2%AB_i_%C2%BB%E2%80%A6_pour_aller_vers_le_lien_entre_cos_sin_et_exp\" >Les puissances de \u00ab\u00a0i\u00a0\u00bb&#8230; pour aller vers le lien entre cos, sin et exp<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/02\/26\/lien-entre-cos-sin-et-exp\/#cosx_isinx\" >cos(x) + isin(x)<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_approximation_par_des_polynomes_pour_connaitre_le_lien_entre_cos_sin_et_exp\"><\/span>Une approximation par des polyn\u00f4mes pour conna\u00eetre le lien entre cos, sin et exp<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_fonction_sinus\"><\/span>La fonction sinus<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Commen\u00e7ons par parler de la fonction sinus; il y a une chose qui est plut\u00f4t marrante : on peut l&rsquo;approximer \u00e0 l&rsquo;aide de polyn\u00f4mes&#8230; Regardez plut\u00f4t:<\/p>\n\n\n\n<figure class=\"wp-block-video aligncenter\"><video height=\"848\" style=\"aspect-ratio: 1600 \/ 848;\" width=\"1600\" controls src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/sinus.mp4\"><\/video><figcaption>Approximation de la fonction sinus<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Au d\u00e9but, on trace la courbe repr\u00e9sentative de la fonction <em>g<\/em> : \\(x \\mapsto x\\), qui colle assez bien \u00e0 la courbe de la fonction sinus sur [-0,5 ; 0,5]. On modifie alors la fonction <em>g<\/em> pour \\(g(x)=x-\\frac{x^3}{3!}\\), o\u00f9 \u00ab\u00a03!\u00a0\u00bb d\u00e9signe la \u00ab\u00a0factorielle de 3\u00a0\u00bb, c&rsquo;est&#8212;dire \\(1\\times2\\times3\\), et on voit que sa repr\u00e9sentation graphique colle un peu mieux \u00e0 celle de la fonction sinus qu&rsquo;auparavant&#8230; On continue alors avec \\( g(x)=x-\\frac{x^3}{3!}+\\frac{x^5}{5!}\\) et \u00e7a colle encore mieux. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">On continue alors sur le m\u00eame sch\u00e9ma et en fait, on peut assez vite se rendre compte que plus on ajoute de termes sur le m\u00eame mod\u00e8le, plus la courbe colle \u00e0 celle de la fonction sinus. On arrive m\u00eame \u00e0 d\u00e9montrer qu&rsquo;\u00e0 l&rsquo;infini, il y a \u00e9galit\u00e9:$$\\sin(x)=\\sum_{n=0}^{+\\infty}\\frac{(-1)^nx^{2n+1}}{(2n+1)!}$$Dans cette somme, il n&rsquo;y a que des puissances impaires de <em>x<\/em>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_fonction_cosinus\"><\/span>La fonction cosinus<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Pour la fonction cosinus, on va faire de m\u00eame en changeant l\u00e9g\u00e8rement le mod\u00e8le:<\/p>\n\n\n\n<figure class=\"wp-block-video aligncenter\"><video height=\"832\" style=\"aspect-ratio: 1584 \/ 832;\" width=\"1584\" controls src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/cosinus.mp4\"><\/video><figcaption>Approximation de la fonction cosinus<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Cette fois ci, les puissances de <em>x<\/em> qui interviennent sont non plus impaires, mais paires (quoi de plus normal dans la mesure o\u00f9 la fonction cosinus est paire alors que la fonction sinus est impaire). On arrive \u00e0 d\u00e9montrer que:$$\\cos(x)=\\sum_{n=0}^{+\\infty}\\frac{(-1)^nx^{2n}}{(2n)!}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_fonction_exponentielle\"><\/span>La fonction exponentielle<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Continuons sur notre lanc\u00e9e avec la fonction exponentielle:<\/p>\n\n\n\n<figure class=\"wp-block-video aligncenter\"><video height=\"832\" style=\"aspect-ratio: 1584 \/ 832;\" width=\"1584\" controls src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/02\/exponentielle.mp4\"><\/video><figcaption>Approximation de la fonction exponentielle<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Cette fois, on peut d\u00e9montrer que:$$\\exp(x)=\\sum_{n=0}^{+\\infty}\\frac{x^n}{n!}$$Attention: les vid\u00e9os ne permettent pas de d\u00e9montrer les trois \u00e9galit\u00e9s; elles ne servent qu&rsquo;\u00e0 visualiser que pour les premi\u00e8res valeurs de <em>n<\/em>, \u00e7a fonctionne. Les d\u00e9monstrations rigoureuses ne peuvent se faire qu&rsquo;une fois un niveau de math\u00e9matiques acceptable acquis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Quoi qu&rsquo;il en soit, nous voyons que la formule sur la fonction exponentielle ressemble aux deux autres, \u00e0 peu de choses pr\u00e8s&#8230;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Les_puissances_de_%C2%AB_i_%C2%BB%E2%80%A6_pour_aller_vers_le_lien_entre_cos_sin_et_exp\"><\/span>Les puissances de \u00ab\u00a0i\u00a0\u00bb&#8230; pour aller vers le lien entre cos, sin et exp<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Vous le savez peut-\u00eatre, un nombre complexe est d\u00e9fini comme \u00e9tant un nombre alg\u00e9brique abstrait qui peut s&rsquo;\u00e9crire sous la forme <em>z<\/em> = <em>x<\/em> + i<em>y<\/em>, avec i\u00b2=-1.  De cette d\u00e9finition, on peut alors \u00e9crire:$$\\text{i}^3=\\text{i}^2 \\times \\text{i}=-1 \\times \\text{i} =-\\text{i}.$$De plus, $$\\text{i}^4=(\\text{i}^2)^2=(-1)^2=1.$$On d\u00e9montre alors que pour tout entier naturel <em>n<\/em>,$$\\text{i}^{4n}=1,\\ \\text{i}^{4n+1}=\\text{i},\\ \\text{i}^{4n+2}=-1,\\ \\text{i}^{4n+3}=-\\text{i}.$$Autrement dit,$$\\text{i}^{2k}=(-1)^k\\text{ et }\\text{i}^{2k+1}=(-1)^k\\text{i}.$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"cosx_isinx\"><\/span>cos(x) + isin(x)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Bon, ben avec tout \u00e7a, on va pouvoir avancer et voir le lien entre cos, sin et exp&#8230;$$\\cos(x)+\\text{i}\\sin(x) =  \\sum_{n=0}^{+\\infty}\\frac{(-1)^nx^{2n}}{(2n)!}  + \\text{i} \\sum_{n=0}^{+\\infty}\\frac{(-1)^nx^{2n+1}}{(2n+1)!}$$et:$$\\begin{align}\\exp(\\text{i}x) &amp; =  \\sum_{n=0}^{+\\infty}\\frac{(\\text{i}x)^n}{n!} \\\\ &amp; =   \\sum_{n=0}^{+\\infty}\\text{i}^n\\frac{x^n}{n!} \\\\ &amp; =   \\sum_{k=0}^{+\\infty}\\text{i}^{2k}\\frac{x^{2k}}{(2k)!}  +  \\sum_{k=0}^{+\\infty}\\text{i}^{2k+1}\\frac{x^{2k+1}}{(2k+1)!}  \\\\ &amp; =  \\sum_{k=0}^{+\\infty}(-1)^k\\frac{x^{2k}}{(2k)!}  +  \\sum_{k=0}^{+\\infty}\\text{i}(-1)^k\\frac{x^{2k+1}}{(2k+1)!}  \\\\ &amp; =  \\sum_{k=0}^{+\\infty}(-1)^k\\frac{x^{2k}}{(2k)!}  +   \\text{i} \\sum_{k=0}^{+\\infty}(-1)^k\\frac{x^{2k+1}}{(2k+1)!} \\\\&amp; = \\cos(x) + \\text{i}\\sin(x).  \\end{align}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Et voil\u00e0 pourquoi on peut se permettre d&rsquo;utiliser une notation exponentielle \u00e0 la place d&rsquo;une notation trigonom\u00e9trique. C&rsquo;est certes plus pratique dans les calculs, mais c&rsquo;est surtout parce que nous avons une parfaite \u00e9galit\u00e9!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">En France, les nombres complexes sont sortis du <a href=\"https:\/\/www.mathweb.fr\/euclide\/exercices-corriges-maths-terminale\/\">programme \u00ab\u00a0standard\u00a0\u00bb<\/a> pour se retrouver enseign\u00e9s en \u00ab\u00a0maths expertes\u00a0\u00bb.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nous allons voir ce qui lit cos, sin et exp. Un nombre complexe admet trois \u00e9critures : sa forme alg\u00e9brique (z = x + iy), sa forme trigonom\u00e9trique (z = r[cos(t) + isin(t)]) et&#8230; sa forme exponentielle (z = exp(it)). Jusqu&rsquo;en 2020, les \u00e9l\u00e8ves de terminale de France voyaient cette [&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],"tags":[178,176,180,179,163,177,150],"class_list":["post-2062","post","type-post","status-publish","format-standard","hentry","category-enseignement","category-mathematiques","tag-cosinus","tag-exponentielle","tag-forme-exponentielle","tag-forme-trigonometrique","tag-nombres-complexes","tag-sinus","tag-trigonometrie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>cos sin exp : pourquoi \u00e9crit-on cos(x)+isin(x)=exp(ix)? - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Quel lien existe-t-il entre cos, sin et exp ? Pourquoi \u00e9crit-on cos(x) + isin(x) = exp(ix) ? 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