{"id":408,"date":"2023-07-04T13:41:37","date_gmt":"2023-07-04T13:41:37","guid":{"rendered":"https:\/\/mathority.org\/nl\/hyperbolische-cosinusfunctie\/"},"modified":"2023-07-04T13:41:37","modified_gmt":"2023-07-04T13:41:37","slug":"hyperbolische-cosinusfunctie","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/hyperbolische-cosinusfunctie\/","title":{"rendered":"Hyperbolische cosinusfunctie"},"content":{"rendered":"<p>Hier vindt u alles over de hyperbolische cosinusfunctie: wat is de formule, de grafische weergave, de kenmerken ervan, de wiskundige relaties met andere functies, enz. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-coseno-hiperbolico\"><\/span> Hyperbolische cosinusformule<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> De <strong>hyperbolische cosinusfunctie<\/strong> is een van de belangrijkste hyperbolische functies en wordt weergegeven door het symbool <strong>cosh(x)<\/strong> . De hyperbolische cosinus is gelijk aan de som van e <sup>x<\/sup> plus e <sup>-x<\/sup> gedeeld door 2.<\/p>\n<p> Daarom is de formule voor de hyperbolische cosinus:<\/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-f667377a7e792f2c9f5f5a7a0ecda4e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{cosh}(x)=\\cfrac{e^{x}+e^{-x}}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"149\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> De hyperbolische cosinus is dus wiskundig gerelateerd aan de exponenti\u00eble functie. In de volgende link kunt u de eigenschappen van dit type functie zien:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zie:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\/exponentiele-functie\/\">eigenschappen van de exponenti\u00eble functie<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"representacion-grafica-del-coseno-hiperbolico\"><\/span> Grafische weergave van de hyperbolische cosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> De grafische weergave van de hyperbolische cosinusfunctie heeft de vorm van een kwadratische functie (of parabool): <\/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\/cosinus-hyperbolique.webp\" alt=\"hyperbolische cosinus\" class=\"wp-image-376\" width=\"281\" height=\"308\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zie:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\/kwadratische-paraboolfunctie\/\">Grafische weergave van een kwadratische functie<\/a><\/span> .<\/p>\n<p> In deze grafiek kunnen we duidelijk zien dat de hyperbolische cosinus een even functie is, omdat deze symmetrisch is rond de y-as.<\/p>\n<p> Aan de andere kant is de grafiek van de hyperbolische cosinus heel anders dan die van de cosinus (trigonometrische functie), die een periodieke functie is. U kunt de grafische weergave van de cosinus en alle verschillen met de hyperbolische cosinus zien in de volgende link:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zie:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\/cosinus-functie\/\">grafische weergave van de cosinusfunctie<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-del-coseno-hiperbolico\"><\/span> Kenmerken van de hyperbolische cosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> De hyperbolische cosinus respecteert de volgende eigenschappen:<\/p>\n<ul>\n<li> Het domein van de hyperbolische cosinusfunctie bestaat uit alle re\u00eble getallen:<\/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> In plaats daarvan is het bereik (of bereik) van de hyperbolische cosinusfunctie 1 en alle getallen groter dan 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-60c76341385ea1ebb5f20476cd8226f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{Im } f= [1,+\\infty)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> De hyperbolische cosinus is een continue en even functie.<\/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-91f3c4459fc8d8840fd902946c851d1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cosh}(-x)=\\text{cosh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> De functie snijdt de Y-as in het punt x=0.<\/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-dd01415f329053c1a450867378fc1582_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Aan de andere kant heeft de functie geen snijpunt met de X-as.<\/li>\n<\/ul>\n<ul>\n<li> De twee grenzen tot oneindig (positief en negatief) van de hyperbolische cosinusfunctie geven plus oneindig.<\/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-5a5aadc0c48684e553e9971aefe442d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\text{cosh}(x)=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"161\" 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-fb2a605c43e81635473abf73554d264f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to-\\infty}\\text{cosh}(x)=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"161\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<ul>\n<li> De hyperbolische cosinus neemt af tot x = 0 en neemt vanaf dat punt oneindig toe, dus de functie heeft een minimum bij x = 0.<\/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-dd01415f329053c1a450867378fc1582_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(0,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> De functie is over het hele domein convex en heeft dus geen buigpunt.<\/li>\n<\/ul>\n<ul>\n<li> De afgeleide van de hyperbolische cosinusfunctie is de hyperbolische sinus:<\/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-9a7a982aae764353843a57652f9a6797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\text{cosh}(x) \\ \\longrightarrow \\ f'(x)=\\text{senh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> De integraal van de hyperbolische cosinusfunctie is de hyperbolische sinus:<\/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-a509c96c515bb7115359764e7e4451df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\int \\text{cosh}(x) \\ dx= \\text{senh}(x) + C\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"221\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<ul>\n<li> Het Taylor-polynoom (of Maclaurin-reeks) van de hyperbolische cosinusfunctie is als volgt:<\/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-cdbe87b63234b62e226371256f8a6c8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{cosh}(x)=1+\\cfrac{x^2}{2!}+\\cfrac{x^4}{4!}+\\cfrac{x^6}{6!}+\\dots=\\sum_{n=0}^\\infty\\cfrac{x^{2n}}{(2n)!}\" title=\"Rendered by QuickLaTeX.com\" height=\"50\" width=\"350\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<ul>\n<li> De Laplace-transformatie van de hyperbolische cosinusfunctie is als volgt: <\/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-ee373a19450972f0c4cccc3a273770e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mathcal{L}\\bigl[\\text{cosh}(at)\\bigr]=\\cfrac{s}{s^2-a^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"171\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"relaciones-matematicas-del-coseno-hiperbolico\"><\/span> Wiskundige relaties van hyperbolische cosinus<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Vervolgens zullen we zien hoe de hyperbolische cosinus kan worden berekend op basis van andere hyperbolische functies, aangezien ze allemaal wiskundig gerelateerd zijn.<\/p>\n<p> De fundamentele vergelijking relateert de hyperbolische cosinus aan de hyperbolische sinus:<\/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> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zie:<\/strong> <a href=\"https:\/\/mathority.org\/nl\/hyperbolische-sinusfunctie\/\"><span style=\"text-decoration: underline;\">hyperbolische sinus<\/span><\/a><\/p>\n<p> De drie belangrijkste hyperbolische functies (hyperbolische sinus, cosinus en tangens) kunnen met elkaar in verband worden gebracht door de volgende vergelijking:<\/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> Aan de andere kant kan de hyperbolische cosinus van het optellen (of aftrekken) van twee verschillende getallen worden bepaald met de volgende formules:<\/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-75dff3fbdfd533e08cf581767a0d9b7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}(x+y)=\\text{cosh}(x)\\text{cosh}(y)+\\text{senh}(y)\\text{senh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"363\" 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-3762e67762eecb02ed3d30c39febca9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}(x-y)=\\text{cosh}(x)\\text{cosh}(y)-\\text{senh}(y)\\text{senh}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"363\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> De hyperbolische cosinus van tweemaal een getal is gelijk aan de som van de kwadraten van de hyperbolische cosinus en de hyperbolische sinus van dit getal:<\/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-be5e2f8a1f407e6dce0c53932575d545_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}(2x)=\\text{cosh}^2(x)+\\text{senh}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"242\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Het optellen of aftrekken van twee hyperbolische cosinussen kan worden berekend door de volgende formules toe te passen:<\/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-02b9d1c798f2639e6503add69dcdb401_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{cosh}(x)+\\text{cosh}(y)=2\\text{cosh}\\left(\\frac{x+y}{2}\\right)\\text{cosh}\\left(\\frac{x-y}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"384\" 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-0ec9d613adc720e383f1c3c0c9c8ca5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{cosh}(x)-\\text{cosh}(y)=2\\text{senh}\\left(\\frac{x+y}{2}\\right)\\text{senh}\\left(\\frac{x-y}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"386\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Ten slotte kan het kwadraat van de hyperbolische cosinus worden berekend met de volgende formule:<\/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-33c90357c8680ebd2ca5725aad7703f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cosh}^2(x)=\\cfrac{1}{2}\\Bigl(\\text{cosh}(2x)+1\\Bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"215\" style=\"vertical-align: -12px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Hier vindt u alles over de hyperbolische cosinusfunctie: wat is de formule, de grafische weergave, de kenmerken ervan, de wiskundige relaties met andere functies, enz. Hyperbolische cosinusformule De hyperbolische cosinusfunctie is een van de belangrijkste hyperbolische functies en wordt weergegeven door het symbool cosh(x) . De hyperbolische cosinus is gelijk aan de som van e &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/hyperbolische-cosinusfunctie\/\"> <span class=\"screen-reader-text\">Hyperbolische cosinusfunctie<\/span> Lees meer &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":[49],"tags":[],"class_list":["post-408","post","type-post","status-publish","format-standard","hentry","category-functie-representatie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Hyperbolische cosinusfunctie - 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\/nl\/hyperbolische-cosinusfunctie\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hyperbolische cosinusfunctie - Mathority\" \/>\n<meta property=\"og:description\" content=\"Hier vindt u alles over de hyperbolische cosinusfunctie: wat is de formule, de grafische weergave, de kenmerken ervan, de wiskundige relaties met andere functies, enz. Hyperbolische cosinusformule De hyperbolische cosinusfunctie is een van de belangrijkste hyperbolische functies en wordt weergegeven door het symbool cosh(x) . 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Hyperbolische cosinusformule De hyperbolische cosinusfunctie is een van de belangrijkste hyperbolische functies en wordt weergegeven door het symbool cosh(x) . 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