{"id":437,"date":"2018-12-24T10:36:14","date_gmt":"2018-12-24T09:36:14","guid":{"rendered":"https:\/\/blogs.ua.es\/matesfacil\/?page_id=437"},"modified":"2019-06-13T08:56:12","modified_gmt":"2019-06-13T07:56:12","slug":"forma-binomica-trigonometrica-y-polar-de-numeros-imaginarios","status":"publish","type":"page","link":"https:\/\/blogs.ua.es\/matesfacil\/secundaria-numeros-operaciones\/numeros-imaginarios\/forma-binomica-trigonometrica-y-polar-de-numeros-imaginarios\/","title":{"rendered":"Forma bin\u00f3mica, trigonom\u00e9trica y polar de n\u00fameros imaginarios"},"content":{"rendered":"<p>En la <strong>forma bin\u00f3mica<\/strong>, un complejo\u00a0<span id=\"MathJax-Element-9-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">z<\/span><\/span>\u00a0se escribe como la suma de un n\u00famero real\u00a0<span id=\"MathJax-Element-10-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a<\/span><\/span>\u00a0y un n\u00famero real\u00a0<span id=\"MathJax-Element-11-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">b<\/span><\/span>\u00a0multiplicado por la unidad imaginaria\u00a0<span id=\"MathJax-Element-12-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">i<\/span><\/span>:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"z = a + b\u00b7i\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/T0.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" width=\"100\" height=\"18\" \/><\/a><\/p>\n<p>El n\u00famero\u00a0<span id=\"MathJax-Element-13-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a<\/span><\/span>\u00a0es la\u00a0<strong>parte real<\/strong>\u00a0de\u00a0<span id=\"MathJax-Element-14-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">z<\/span><\/span>\u00a0y\u00a0<span id=\"MathJax-Element-15-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">b<\/span><\/span>\u00a0es la\u00a0<strong>parte imaginaria<\/strong>\u00a0de\u00a0<span id=\"MathJax-Element-16-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">z<\/span><\/span>.<\/p>\n<p>&nbsp;<\/p>\n<p>La forma <strong>trigonom\u00e9trica<\/strong> del complejo\u00a0<span id=\"MathJax-Element-24-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-143\" class=\"mjx-math\"><span id=\"MJXc-Node-144\" class=\"mjx-mrow\"><span id=\"MJXc-Node-145\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><span id=\"MJXc-Node-146\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-147\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-148\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-149\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-150\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">i\u00a0<\/span><\/span><\/span><\/span><\/span>\u00a0es<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"f\u00f3rmula trigonom\u00e9trica de un n\u00famero complejo: z = |z|\u00b7(cos(\u03b1)+i\u00b7sin(\u03b1))\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/T3.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" width=\"240\" height=\"144\" \/><\/a><\/p>\n<p>El \u00e1ngulo \u03b1 que proporciona la funci\u00f3n arcotangente es siempre entre -45\u00b0 y 45\u00b0. Si el complejo pertenece el primer cuadrante (<span id=\"MathJax-Element-25-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-151\" class=\"mjx-math\"><span id=\"MJXc-Node-152\" class=\"mjx-mrow\"><span id=\"MJXc-Node-153\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-154\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-155\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>,\u00a0<span id=\"MathJax-Element-26-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-156\" class=\"mjx-math\"><span id=\"MJXc-Node-157\" class=\"mjx-mrow\"><span id=\"MJXc-Node-158\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-159\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-160\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>) o al cuarto (<span id=\"MathJax-Element-27-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-161\" class=\"mjx-math\"><span id=\"MJXc-Node-162\" class=\"mjx-mrow\"><span id=\"MJXc-Node-163\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-164\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-165\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>,\u00a0<span id=\"MathJax-Element-28-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-166\" class=\"mjx-math\"><span id=\"MJXc-Node-167\" class=\"mjx-mrow\"><span id=\"MJXc-Node-168\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-169\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-170\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>), el \u00e1ngulo obtenido es el argumento del complejo.<\/p>\n<p>Sin embargo, si el complejo est\u00e1 en el segundo cuadrante (<span id=\"MathJax-Element-29-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-171\" class=\"mjx-math\"><span id=\"MJXc-Node-172\" class=\"mjx-mrow\"><span id=\"MJXc-Node-173\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-174\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-175\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>,\u00a0<span id=\"MathJax-Element-30-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-176\" class=\"mjx-math\"><span id=\"MJXc-Node-177\" class=\"mjx-mrow\"><span id=\"MJXc-Node-178\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-179\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-180\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>), hay que sumarle 180\u00b0. Y si est\u00e1 en el tercer cuadrante, (<span id=\"MathJax-Element-31-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-181\" class=\"mjx-math\"><span id=\"MJXc-Node-182\" class=\"mjx-mrow\"><span id=\"MJXc-Node-183\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-184\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-185\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>,\u00a0<span id=\"MathJax-Element-32-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-186\" class=\"mjx-math\"><span id=\"MJXc-Node-187\" class=\"mjx-mrow\"><span id=\"MJXc-Node-188\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-189\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-190\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>), hay que restarle 180\u00b0.<\/p>\n<p>Hay una funci\u00f3n proporciona directamente el argumento:\u00a0<span id=\"MathJax-Element-33-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-191\" class=\"mjx-math\"><span id=\"MJXc-Node-192\" class=\"mjx-mrow\"><span id=\"MJXc-Node-193\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-194\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">t<\/span><\/span><span id=\"MJXc-Node-195\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-196\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">n<\/span><\/span><span id=\"MJXc-Node-197\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><span id=\"MJXc-Node-198\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-199\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-200\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-201\" class=\"mjx-mi MJXc-space1\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-202\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span>.<\/p>\n<p><span id=\"MathJax-Element-39-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 16.432px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-233\" class=\"mjx-math\"><span id=\"MJXc-Node-234\" class=\"mjx-mrow\"><span id=\"MJXc-Node-239\" class=\"mjx-texatom\"><span id=\"MJXc-Node-240\" class=\"mjx-mrow\"><span id=\"MJXc-Node-241\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">El m\u00f3dulo,\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" style=\"font-size: 16.432px\" role=\"presentation\">|z|,<\/span><\/span>\u00a0es la ra\u00edz cuadrada de la suma del cuadrado de la parte real y de la parte imaginaria:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el m\u00f3dulo de z = a+bi es |z| = ra\u00edz(a^2 + b^2)\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/T1.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" width=\"139\" height=\"27\" \/><\/a><\/p>\n<p><strong>Ejemplo:\u00a0<\/strong>escribimos el complejo z = 1-i en forma trigonom\u00e9trica:<\/p>\n<p>Calculamos el m\u00f3dulo:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el m\u00f3dulo de 1-i es ra\u00edz(2)\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/P2-1.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" width=\"183\" height=\"55\" \/><\/a><\/p>\n<p>Calculamos el \u00e1ngulo que forma\u00a0<span id=\"MathJax-Element-49-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 16.432px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-304\" class=\"mjx-math\"><span id=\"MJXc-Node-305\" class=\"mjx-mrow\"><span id=\"MJXc-Node-306\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z:<\/span><\/span><\/span><\/span><\/span><\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el argumento de 1-i es -45\u00ba\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/P2-2.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" width=\"160\" height=\"70\" \/><\/a><\/p>\n<p>Por tanto, la forma trigonom\u00e9trica de\u00a0<span id=\"MathJax-Element-50-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 16.432px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-307\" class=\"mjx-math\"><span id=\"MJXc-Node-308\" class=\"mjx-mrow\"><span id=\"MJXc-Node-309\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><\/span><\/span><\/span>\u00a0es<\/p>\n<p align=\"center\"><img decoding=\"async\" title=\"la forma trigonom\u00e9trica del complejo 1-i es ra\u00edz(2)\u00b7(cos(-45\u00ba)+i\u00b7sin(-45\u00ba))\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/P2-3.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" \/><\/p>\n<p>En la <strong>forma polar<\/strong>, el complejo se escribe en funci\u00f3n de su m\u00f3dulo\u00a0<span id=\"MathJax-Element-70-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-412\" class=\"mjx-math\"><span id=\"MJXc-Node-413\" class=\"mjx-mrow\"><span id=\"MJXc-Node-414\" class=\"mjx-texatom\"><span id=\"MJXc-Node-415\" class=\"mjx-mrow\"><span id=\"MJXc-Node-416\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">|<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-417\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><span id=\"MJXc-Node-418\" class=\"mjx-texatom\"><span id=\"MJXc-Node-419\" class=\"mjx-mrow\"><span id=\"MJXc-Node-420\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">|<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0y su argumento\u00a0<span id=\"MathJax-Element-71-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-421\" class=\"mjx-math\"><span id=\"MJXc-Node-422\" class=\"mjx-mrow\"><span id=\"MJXc-Node-423\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03b1\u00a0<\/span><\/span><\/span><\/span><\/span>como<\/p>\n<p align=\"center\"><img decoding=\"async\" title=\"forma polar: z = |z|\u00b7e^(i\u00b7\u03b1)\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/T4.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" \/><\/p>\n<p style=\"text-align: left\" align=\"center\"><strong>Ejemplo:\u00a0<\/strong>escribimos el n\u00famero imaginario z = -1+i en forma polar<\/p>\n<p>Calculamos el m\u00f3dulo del complejo\u00a0<span id=\"MathJax-Element-73-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 16.432px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">z<\/span><\/span>:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el m\u00f3dulo de -1+i es ra\u00edz(2)\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/P5-1.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" width=\"176\" height=\"60\" \/><\/a><\/p>\n<p>Calculamos su argumento:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el argumento de -1+i es 135\u00ba\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/P5-2.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" width=\"162\" height=\"99\" \/><\/a><\/p>\n<p><strong>Nota:<\/strong>\u00a0hemos sumado 180\u00ba grados al \u00e1ngulo obtenido (-45\u00ba) porque el complejo est\u00e1 en el segundo cuadrante).<\/p>\n<p>Por tanto, la forma polar de\u00a0<span id=\"MathJax-Element-74-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 16.432px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-434\" class=\"mjx-math\"><span id=\"MJXc-Node-435\" class=\"mjx-mrow\"><span id=\"MJXc-Node-436\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><\/span><\/span><\/span>\u00a0es<\/p>\n<p align=\"center\"><img decoding=\"async\" title=\"la forma polar del complejo -1+i es ra\u00edz(2)\u00b7e^(135\u00ba\u00b7i)\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img2\/P5-3.png\" alt=\"Formas bin\u00f3mica, trigonom\u00e9trica y polar de los n\u00fameros complejos o imaginarios. Con ejemplos, problemas resueltos y representaciones. Secundaria, Bachillerato y Universidad.\" \/><\/p>\n<p style=\"text-align: left\">M\u00e1s informaci\u00f3n:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-definicion-representacion-raiz-negativos-i.html\">Introducci\u00f3n a los n\u00fameros complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\">Formas bin\u00f3mica y polar<\/a><\/li>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-modulo-argumento-angulo-propiedades.html\">M\u00f3dulo y argumento de complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/sumar-restar-multiplicar-dividir-numeros-complejos-imaginarios-ejemplos-formulas.html\">Operaciones entre complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/ejercicios-resueltos-producto-complejos.html\">Producto y cociente de complejos en forma bin\u00f3mica<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/BAC\/complejos\/numeros-complejos-forma-polar-binomica-calculadora-producto-problemas-resueltos.html\">Producto y cociente de complejos en forma polar<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/ejercicios-resueltos-demostraciones-complejos.html\">Propiedades de los n\u00fameros complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/BAC\/complejos\/raices\/raices-n-esimas-numeros-complejos-imaginarios-poligono-regular-argumento-modulo-ejemplos.html\">Ra\u00edces de n\u00fameros complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/ejercicios-resueltos-producto-complejos.html\">Calculadora de operaciones entre complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/BAC\/complejos\/numeros-complejos-forma-polar-binomica-calculadora-producto-problemas-resueltos.html\">Calculadora de forma polar y bin\u00f3mica de complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/SegundoGrado\/ECUACIONES-SEGUNDO-GRADO-SOLUCIONES-COMPLEJAS.html\">Ecuaciones cuadr\u00e1ticas con soluciones complejas<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>En la forma bin\u00f3mica, un complejo\u00a0z\u00a0se escribe como la suma de un n\u00famero real\u00a0a\u00a0y un n\u00famero real\u00a0b\u00a0multiplicado por la unidad imaginaria\u00a0i: El n\u00famero\u00a0a\u00a0es la\u00a0parte real\u00a0de\u00a0z\u00a0y\u00a0b\u00a0es la\u00a0parte imaginaria\u00a0de\u00a0z. &nbsp; La forma trigonom\u00e9trica del complejo\u00a0z=a+bi\u00a0\u00a0es El \u00e1ngulo \u03b1 que proporciona la funci\u00f3n arcotangente es siempre entre -45\u00b0 y 45\u00b0. Si el complejo pertenece el primer cuadrante (a&gt;0,\u00a0b&gt;0) &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/blogs.ua.es\/matesfacil\/secundaria-numeros-operaciones\/numeros-imaginarios\/forma-binomica-trigonometrica-y-polar-de-numeros-imaginarios\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Forma bin\u00f3mica, trigonom\u00e9trica y polar de n\u00fameros imaginarios&#8221;<\/span><\/a><\/p>\n","protected":false},"author":4324,"featured_media":0,"parent":412,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-437","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/437","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/users\/4324"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/comments?post=437"}],"version-history":[{"count":3,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/437\/revisions"}],"predecessor-version":[{"id":617,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/437\/revisions\/617"}],"up":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/412"}],"wp:attachment":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/media?parent=437"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}