{"id":364,"date":"2023-07-04T12:37:32","date_gmt":"2023-07-04T12:37:32","guid":{"rendered":"https:\/\/mathority.org\/pt\/funcao-tangente-hiperbolica\/"},"modified":"2023-07-04T12:37:32","modified_gmt":"2023-07-04T12:37:32","slug":"funcao-tangente-hiperbolica","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/funcao-tangente-hiperbolica\/","title":{"rendered":"Fun\u00e7\u00e3o tangente hiperb\u00f3lica"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 tudo sobre a tangente hiperb\u00f3lica: qual a sua f\u00f3rmula, sua representa\u00e7\u00e3o gr\u00e1fica, todas as suas caracter\u00edsticas,\u2026 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-tangente-hiperbolica\"><\/span> F\u00f3rmula tangente hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A fun\u00e7\u00e3o <strong>tangente hiperb\u00f3lica<\/strong> \u00e9 uma das principais fun\u00e7\u00f5es hiperb\u00f3licas e \u00e9 representada pelo s\u00edmbolo <strong>tanh(x)<\/strong> . Matematicamente, a tangente hiperb\u00f3lica \u00e9 igual ao seno hiperb\u00f3lico dividido pelo cosseno hiperb\u00f3lico.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12f286528bc0635705aadbe510b6ceb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}(x)=\\cfrac{\\text{senh}(x)}{\\text{cosh}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"144\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> A partir da <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/funcao-seno-hiperbolica\/\">f\u00f3rmula do seno hiperb\u00f3lico<\/a><\/span> e da <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/funcao-cosseno-hiperbolica\/\">f\u00f3rmula do cosseno hiperb\u00f3lico,<\/a><\/span> podemos chegar \u00e0 seguinte express\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a53ac0ed7df921993e36d27fdcda71c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}(x)=\\cfrac{e^x-e^{-x}}{e^x+e^{-x}}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"151\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> Portanto, a fun\u00e7\u00e3o tangente hiperb\u00f3lica est\u00e1 relacionada \u00e0 fun\u00e7\u00e3o exponencial. No link a seguir voc\u00ea pode ver todas as caracter\u00edsticas desses tipos de fun\u00e7\u00f5es:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/funcao-exponencial\/\">caracter\u00edsticas das fun\u00e7\u00f5es exponenciais<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"representacion-grafica-de-la-tangente-hiperbolica\"><\/span> Representa\u00e7\u00e3o gr\u00e1fica da tangente hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A partir de sua f\u00f3rmula, podemos representar graficamente a fun\u00e7\u00e3o tangente hiperb\u00f3lica: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/tangente-hyperbolique.webp\" alt=\"tangente hiperb\u00f3lica\" class=\"wp-image-403\" width=\"349\" height=\"276\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Como voc\u00ea pode ver no gr\u00e1fico, a fun\u00e7\u00e3o tangente hiperb\u00f3lica tem duas ass\u00edntotas horizontais em x=+1 e x=-1, uma vez que o limite da fun\u00e7\u00e3o quando x se aproxima de mais infinito d\u00e1 x=+1, e o limite para menos infinito d\u00e1 x=-1.<\/p>\n<p> Por outro lado, o gr\u00e1fico da tangente hiperb\u00f3lica nada tem a ver com o gr\u00e1fico da tangente (fun\u00e7\u00e3o trigonom\u00e9trica), que \u00e9 uma fun\u00e7\u00e3o peri\u00f3dica. Voc\u00ea pode ver a representa\u00e7\u00e3o gr\u00e1fica da tangente e como ela difere da tangente hiperb\u00f3lica no seguinte link:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <a href=\"https:\/\/mathority.org\/pt\/funcao-tangente\/\"><span style=\"text-decoration: underline;\">representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o tangente<\/span><\/a> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-de-la-tangente-hiperbolica\"><\/span> Caracter\u00edsticas da tangente hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A fun\u00e7\u00e3o tangente hiperb\u00f3lica tem as seguintes propriedades:<\/p>\n<ul>\n<li> O dom\u00ednio da fun\u00e7\u00e3o tangente hiperb\u00f3lica s\u00e3o todos os n\u00fameros reais.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0cd1539b66edeb38040ed80168e1fd9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Dom } f = \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<ul>\n<li> Em contraste, o caminho ou intervalo da fun\u00e7\u00e3o tangente hiperb\u00f3lica \u00e9 limitado a valores entre -1 e +1 (n\u00e3o inclusivo).<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46fa688a38d3c0a9fed447bd46cd6857_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Im } f= (-1,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> A tangente hiperb\u00f3lica \u00e9 uma fun\u00e7\u00e3o cont\u00ednua, bijetiva e \u00edmpar (sim\u00e9trica em rela\u00e7\u00e3o \u00e0 origem das coordenadas).<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4905247e8dd5f9d0116452745122d04b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tanh}(-x) =- \\text{tanh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"169\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> A fun\u00e7\u00e3o cruza o eixo X e o eixo Y na origem da coordenada.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9cf2000c782cfe94be6df5f499cd3e24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Os limites para mais\/menos infinito da fun\u00e7\u00e3o tangente hiperb\u00f3lica d\u00e3o +1\/-1. Portanto, a fun\u00e7\u00e3o tem uma ass\u00edntota horizontal em x=+1 e outra ass\u00edntota horizontal em x=-1.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-efa518f1c75b0628fee415414c4ddadd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\text{tanh}(x)=+1\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"154\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ccb67d43c129867f0f8d277701221620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to-\\infty}\\text{tanh}(x)=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"154\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<ul>\n<li> A tangente hiperb\u00f3lica \u00e9 estritamente crescente em todo o seu dom\u00ednio, portanto n\u00e3o possui extremos relativos (nem m\u00e1ximo nem m\u00ednimo).<\/li>\n<\/ul>\n<ul>\n<li> No entanto, a fun\u00e7\u00e3o muda de convexa para c\u00f4ncava no ponto x = 0, ent\u00e3o x = 0 \u00e9 um ponto de inflex\u00e3o da fun\u00e7\u00e3o.<\/li>\n<\/ul>\n<ul>\n<li> O inverso da fun\u00e7\u00e3o tangente hiperb\u00f3lica \u00e9 chamado de argumento da tangente hiperb\u00f3lica (ou arco tangente hiperb\u00f3lico) e sua f\u00f3rmula \u00e9 a seguinte:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8258540cca67218d148d2599727d907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{tanh}^{-1}(x)=\\text{arg tanh}(x)=\\cfrac{1}{2}\\ln\\left(\\frac{1+x}{1-x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"314\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul>\n<li> A derivada da fun\u00e7\u00e3o tangente hiperb\u00f3lica \u00e9 1 dividido pelo quadrado do cosseno hiperb\u00f3lico:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2ad0f1a0c4fd6c882bfcdd08f8506c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{tanh}(x) \\ \\longrightarrow \\ f'(x)=\\cfrac{1}{\\text{cosh}^2(x)}=1-\\text{tanh}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"418\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<ul>\n<li> A integral da fun\u00e7\u00e3o tangente hiperb\u00f3lica \u00e9 o logaritmo natural do cosseno hiperb\u00f3lico:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2709f2a36bdbb4b252b040c61bac1309_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\int\\text{tanh}(x) \\ dx= \\ln\\Bigl(\\text{cosh}(x)\\Bigr)+C\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"258\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<ul>\n<li> A tangente hiperb\u00f3lica da soma de dois n\u00fameros diferentes pode ser calculada aplicando a seguinte equa\u00e7\u00e3o:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0291100dea0b530852aa2515f1068f1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tanh}(x+y)=\\cfrac{\\text{tanh}(x)+\\text{tanh}(y)}{1+\\text{tanh}(x)\\cdot \\text{tanh}(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"278\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<ul>\n<li> O polin\u00f4mio de Taylor ou a s\u00e9rie tangente hiperb\u00f3lica tem o raio de converg\u00eancia\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab4119d73bfd1bc300545aa64addcbc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left|x\\right|<\\cfrac{\\pi}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"55\" style=\"vertical-align: -12px;\"><\/p>\n<p> e corresponde \u00e0 seguinte express\u00e3o:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8f0d05ddc7f9bc94f576b83e1c6c88e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{tanh}(x)=x-\\frac{x^3}{3}+\\frac{2x^5}{15}-\\frac{17x^7}{315}+\\cdots =\\sum_{n=1}^\\infty\\frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"515\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p> Ouro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e2075b7c578253ce28ea159b37e5b41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B_n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\"><\/p>\n<p> \u00e9 o <a href=\"https:\/\/es.wikipedia.org\/wiki\/N%C3%BAmero_de_Bernoulli\" target=\"_blank\" rel=\"noreferrer noopener\">n\u00famero de Bernoulli<\/a> .<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 tudo sobre a tangente hiperb\u00f3lica: qual a sua f\u00f3rmula, sua representa\u00e7\u00e3o gr\u00e1fica, todas as suas caracter\u00edsticas,\u2026 F\u00f3rmula tangente hiperb\u00f3lica A fun\u00e7\u00e3o tangente hiperb\u00f3lica \u00e9 uma das principais fun\u00e7\u00f5es hiperb\u00f3licas e \u00e9 representada pelo s\u00edmbolo tanh(x) . Matematicamente, a tangente hiperb\u00f3lica \u00e9 igual ao seno hiperb\u00f3lico dividido pelo cosseno hiperb\u00f3lico. A partir &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/funcao-tangente-hiperbolica\/\"> <span class=\"screen-reader-text\">Fun\u00e7\u00e3o tangente hiperb\u00f3lica<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[22],"tags":[],"class_list":["post-364","post","type-post","status-publish","format-standard","hentry","category-representacao-de-funcao"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Fun\u00e7\u00e3o tangente hiperb\u00f3lica - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/funcao-tangente-hiperbolica\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fun\u00e7\u00e3o tangente hiperb\u00f3lica - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 tudo sobre a tangente hiperb\u00f3lica: qual a sua f\u00f3rmula, sua representa\u00e7\u00e3o gr\u00e1fica, todas as suas caracter\u00edsticas,\u2026 F\u00f3rmula tangente hiperb\u00f3lica A fun\u00e7\u00e3o tangente hiperb\u00f3lica \u00e9 uma das principais fun\u00e7\u00f5es hiperb\u00f3licas e \u00e9 representada pelo s\u00edmbolo tanh(x) . Matematicamente, a tangente hiperb\u00f3lica \u00e9 igual ao seno hiperb\u00f3lico dividido pelo cosseno hiperb\u00f3lico. 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