{"id":362,"date":"2023-07-04T13:53:47","date_gmt":"2023-07-04T13:53:47","guid":{"rendered":"https:\/\/mathority.org\/pt\/funcao-seno-hiperbolica\/"},"modified":"2023-07-04T13:53:47","modified_gmt":"2023-07-04T13:53:47","slug":"funcao-seno-hiperbolica","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/funcao-seno-hiperbolica\/","title":{"rendered":"Fun\u00e7\u00e3o seno hiperb\u00f3lica"},"content":{"rendered":"<p>Neste artigo voc\u00ea encontrar\u00e1 tudo sobre o seno hiperb\u00f3lico: qual a sua f\u00f3rmula, sua representa\u00e7\u00e3o gr\u00e1fica, todas as suas caracter\u00edsticas, as rela\u00e7\u00f5es com outras fun\u00e7\u00f5es,\u2026 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-seno-hiperbolico\"><\/span> F\u00f3rmula seno hiperb\u00f3lica<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A fun\u00e7\u00e3o <strong>seno hiperb\u00f3lica<\/strong> \u00e9 uma das principais fun\u00e7\u00f5es hiperb\u00f3licas e \u00e9 representada pelo s\u00edmbolo <strong>sinh(x)<\/strong> ou <strong>sinh(x)<\/strong> . O seno hiperb\u00f3lico \u00e9 igual a e <sup>x<\/sup> menos e <sup>-x<\/sup> dividido por 2.<\/p>\n<p> A f\u00f3rmula para o seno hiperb\u00f3lico \u00e9, portanto, a seguinte:<\/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-924551c3d4c8b1d68b0c9a7cf2115ca8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{senh}(x)=\\cfrac{e^{x}-e^{-x}}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"150\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Assim, o seno hiperb\u00f3lico est\u00e1 relacionado \u00e0 fun\u00e7\u00e3o exponencial.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <a href=\"https:\/\/mathority.org\/pt\/funcao-exponencial\/\"><span style=\"text-decoration: underline;\">caracter\u00edsticas da fun\u00e7\u00e3o exponencial<\/span><\/a> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"representacion-grafica-del-seno-hiperbolico\"><\/span> Representa\u00e7\u00e3o gr\u00e1fica do seno hiperb\u00f3lico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Usando a f\u00f3rmula que vimos na se\u00e7\u00e3o anterior, podemos fazer uma tabela dos valores dos senos hiperb\u00f3licos e representar graficamente a fun\u00e7\u00e3o: <\/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\/sinus-hyperbolique.webp\" alt=\"seno hiperb\u00f3lico\" class=\"wp-image-346\" width=\"281\" height=\"349\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Neste gr\u00e1fico, podemos ver que o seno hiperb\u00f3lico \u00e9 uma fun\u00e7\u00e3o <strong>\u00edmpar<\/strong> , porque os x opostos t\u00eam imagens opostas, ou seja, o gr\u00e1fico do seno hiperb\u00f3lico \u00e9 sim\u00e9trico em rela\u00e7\u00e3o \u00e0 origem das coordenadas (0, 0).<\/p>\n<p> Como voc\u00ea pode ver, o gr\u00e1fico do seno hiperb\u00f3lico \u00e9 muito diferente daquele do seno, que \u00e9 uma fun\u00e7\u00e3o peri\u00f3dica. Voc\u00ea pode ver a representa\u00e7\u00e3o gr\u00e1fica do seno e todas as diferen\u00e7as com o seno hiperb\u00f3lico no seguinte link:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/funcao-seno\/\">Representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o seno<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-del-seno-hiperbolico\"><\/span> Caracter\u00edsticas do seno hiperb\u00f3lico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O seno hiperb\u00f3lico tem as seguintes propriedades:<\/p>\n<ul>\n<li> O dom\u00ednio da fun\u00e7\u00e3o seno 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> O intervalo ou intervalo da fun\u00e7\u00e3o seno hiperb\u00f3lica tamb\u00e9m s\u00e3o todos 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-5a954b5c192478c3b7b14428ac8d5cbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Im } f= \\mathbb{R}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<ul>\n<li> O seno hiperb\u00f3lico \u00e9 uma fun\u00e7\u00e3o cont\u00ednua e \u00edmpar.<\/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-5b37f1a110d810d8dbf0b5a52a1175c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{senh}(-x) =- \\text{senh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Intercepta o eixo X e o eixo Y no mesmo ponto de interse\u00e7\u00e3o, a 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> O limite da fun\u00e7\u00e3o seno hiperb\u00f3lica quando x tende para mais\/menos infinito \u00e9 igual a mais\/menos infinito:<\/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-14c1cd1cfcf12880c94a1a992fca6636_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\text{senh}(x)=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"162\" 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-d3bad505c476b08f8494c9d4a558ca46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to-\\infty}\\text{senh}(x)=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"162\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<ul>\n<li> O seno hiperb\u00f3lico \u00e9 estritamente crescente em todo o dom\u00ednio, portanto n\u00e3o tem m\u00e1ximos nem m\u00ednimos.<\/li>\n<\/ul>\n<ul>\n<li> Por\u00e9m, ela muda sua curvatura no ponto x = 0, portanto \u00e9 um ponto de inflex\u00e3o da fun\u00e7\u00e3o. Para valores menores que x=0 \u00e9 uma fun\u00e7\u00e3o c\u00f4ncava, por outro lado para valores maiores que x=0 \u00e9 uma fun\u00e7\u00e3o convexa.<\/li>\n<\/ul>\n<ul>\n<li> A derivada da fun\u00e7\u00e3o seno hiperb\u00f3lica \u00e9 o 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-c47c54bd2abef41b1660f7779cd51cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{senh}(x) \\ \\longrightarrow \\ f'(x)=\\text{cosh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Da mesma forma, a integral da fun\u00e7\u00e3o seno hiperb\u00f3lica \u00e9 o 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-fb5ef4d6ad0649cda51f73b558814b45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\int \\text{senh}(x) \\ dx= \\text{cosh}(x) + C\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"221\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<ul>\n<li> A s\u00e9rie de Taylor da fun\u00e7\u00e3o seno hiperb\u00f3lica \u00e9 equivalente \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-3db771459579b232c860bb1276f1d665_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{senh}(x)=x+\\cfrac{x^3}{3!}+\\cfrac{x^5}{5!}+\\cfrac{x^7}{7!}+\\dots=\\sum_{n=0}^\\infty\\cfrac{x^{2n+1}}{(2n+1)!}\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"382\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<ul>\n<li> A transformada de Laplace da fun\u00e7\u00e3o seno hiperb\u00f3lica \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-e3e6b1c1d82a639269c25218155f4019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathcal{L}\\bigl[\\text{senh}(at)\\bigr]=\\cfrac{a}{s^2-a^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"172\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"relaciones-matematicas-del-seno-hiperbolico\"><\/span> Rela\u00e7\u00f5es matem\u00e1ticas do seno hiperb\u00f3lico<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O seno hiperb\u00f3lico est\u00e1 ligado \u00e0s outras fun\u00e7\u00f5es hiperb\u00f3licas pelas seguintes equa\u00e7\u00f5es:<\/p>\n<p> A equa\u00e7\u00e3o fundamental relaciona o seno hiperb\u00f3lico ao 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-4317a445a90e4d139b47db7cf4a49a1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}^2(x)-\\text{senh}^2(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, as fun\u00e7\u00f5es hiperb\u00f3licas seno e cosseno est\u00e3o relacionadas pela equa\u00e7\u00e3o da hip\u00e9rbole, que \u00e9 x <sup>2<\/sup> -y <sup>2<\/sup> =1. Ao contr\u00e1rio das fun\u00e7\u00f5es trigonom\u00e9tricas seno e cosseno que est\u00e3o ligadas pela equa\u00e7\u00e3o do c\u00edrculo (x <sup>2<\/sup> +y <sup>2<\/sup> =1).<\/p>\n<p> As fun\u00e7\u00f5es hiperb\u00f3licas de seno, cosseno e tangente podem ser relacionadas pela seguinte equa\u00e7\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-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> Por outro lado, o seno hiperb\u00f3lico da adi\u00e7\u00e3o ou subtra\u00e7\u00e3o de dois n\u00fameros diferentes pode ser calculado com as seguintes f\u00f3rmulas:<\/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-d0b2492cbc95b27046485a6d7d9cd08c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{senh}(x+y)=\\text{senh}(x)\\text{cosh}(y)+\\text{senh}(y)\\text{cosh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"364\" style=\"vertical-align: -5px;\"><\/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-9a2205899042608a33bd61b158526026_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{senh}(x-y)=\\text{senh}(x)\\text{cosh}(y)-\\text{senh}(y)\\text{cosh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"364\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> O seno hiperb\u00f3lico de duas vezes um n\u00famero pode ser determinado aplicando a seguinte rela\u00e7\u00e3o matem\u00e1tica:<\/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-f0d5c63a53de957eac1f69a4fb209f96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{senh}(2x)=2\\text{senh}(x)\\text{cosh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"215\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A soma ou subtra\u00e7\u00e3o de dois senos hiperb\u00f3licos pode ser encontrada usando as seguintes f\u00f3rmulas:<\/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-9d3598ef49dbf33b925a73edc773a85d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{senh}(x)+\\text{senh}(y)=2\\text{senh}\\left(\\frac{x+y}{2}\\right)\\text{cosh}\\left(\\frac{x-y}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"387\" style=\"vertical-align: -17px;\"><\/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-00ee545f4866d13406bc3d5de7de528a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{senh}(x)-\\text{senh}(y)=2\\text{senh}\\left(\\frac{x-y}{2}\\right)\\text{cosh}\\left(\\frac{x+y}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Finalmente, o quadrado do seno hiperb\u00f3lico pode ser calculado aplicando a seguinte f\u00f3rmula:<\/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-2e3e8284b95e317f876a4f8802ebf5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{senh}^2(x)=\\cfrac{1}{2}\\Bigl(\\text{cosh}(2x)-1\\Bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"216\" style=\"vertical-align: -12px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Neste artigo voc\u00ea encontrar\u00e1 tudo sobre o seno hiperb\u00f3lico: qual a sua f\u00f3rmula, sua representa\u00e7\u00e3o gr\u00e1fica, todas as suas caracter\u00edsticas, as rela\u00e7\u00f5es com outras fun\u00e7\u00f5es,\u2026 F\u00f3rmula seno hiperb\u00f3lica A fun\u00e7\u00e3o seno hiperb\u00f3lica \u00e9 uma das principais fun\u00e7\u00f5es hiperb\u00f3licas e \u00e9 representada pelo s\u00edmbolo sinh(x) ou sinh(x) . O seno hiperb\u00f3lico \u00e9 igual a e x &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/funcao-seno-hiperbolica\/\"> <span class=\"screen-reader-text\">Fun\u00e7\u00e3o seno 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-362","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 seno 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-seno-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 seno hiperb\u00f3lica - Mathority\" \/>\n<meta property=\"og:description\" content=\"Neste artigo voc\u00ea encontrar\u00e1 tudo sobre o seno hiperb\u00f3lico: qual a sua f\u00f3rmula, sua representa\u00e7\u00e3o gr\u00e1fica, todas as suas caracter\u00edsticas, as rela\u00e7\u00f5es com outras fun\u00e7\u00f5es,\u2026 F\u00f3rmula seno hiperb\u00f3lica A fun\u00e7\u00e3o seno hiperb\u00f3lica \u00e9 uma das principais fun\u00e7\u00f5es hiperb\u00f3licas e \u00e9 representada pelo s\u00edmbolo sinh(x) ou sinh(x) . 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