{"id":364,"date":"2023-07-04T13:41:37","date_gmt":"2023-07-04T13:41:37","guid":{"rendered":"https:\/\/mathority.org\/id\/fungsi-kosinus-hiperbolik\/"},"modified":"2023-07-04T13:41:37","modified_gmt":"2023-07-04T13:41:37","slug":"fungsi-kosinus-hiperbolik","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/fungsi-kosinus-hiperbolik\/","title":{"rendered":"Fungsi kosinus hiperbolik"},"content":{"rendered":"<p>Di sini Anda akan menemukan segala sesuatu tentang fungsi kosinus hiperbolik: apa rumusnya, representasi grafisnya, karakteristiknya, hubungan matematisnya dengan fungsi lain, dll. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-coseno-hiperbolico\"><\/span> Rumus kosinus hiperbolik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Fungsi <strong>kosinus hiperbolik<\/strong> adalah salah satu fungsi hiperbolik utama dan diwakili oleh simbol <strong>cosh(x)<\/strong> . Kosinus hiperbolik sama dengan jumlah e <sup>x<\/sup> ditambah e <sup>-x<\/sup> dibagi 2.<\/p>\n<p> Oleh karena itu, rumus kosinus hiperbolik adalah:<\/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> Jadi, kosinus hiperbolik secara matematis berhubungan dengan fungsi eksponensial. Di tautan berikut Anda dapat melihat properti dari jenis fungsi ini:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Lihat:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/fungsi-eksponensial\/\">sifat-sifat fungsi eksponensial<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"representacion-grafica-del-coseno-hiperbolico\"><\/span> Representasi grafis dari kosinus hiperbolik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Representasi grafis dari fungsi kosinus hiperbolik berbentuk fungsi kuadrat (atau parabola): <\/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=\"kosinus hiperbolik\" class=\"wp-image-376\" width=\"281\" height=\"308\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Lihat:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/fungsi-parabola-kuadrat\/\">Representasi grafis dari fungsi kuadrat<\/a><\/span> .<\/p>\n<p> Pada grafik ini terlihat jelas bahwa kosinus hiperbolik merupakan fungsi genap karena simetris terhadap sumbu y.<\/p>\n<p> Sebaliknya, grafik kosinus hiperbolik sangat berbeda dengan grafik kosinus (fungsi trigonometri) yang merupakan fungsi periodik. Gambaran grafis cosinus dan segala perbedaannya dengan cosinus hiperbolik dapat Anda lihat pada link berikut:<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Lihat:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/fungsi-kosinus\/\">representasi grafis dari fungsi kosinus<\/a><\/span> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-del-coseno-hiperbolico\"><\/span> Ciri-ciri kosinus hiperbolik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Kosinus hiperbolik memperhatikan sifat-sifat berikut:<\/p>\n<ul>\n<li> Domain fungsi kosinus hiperbolik adalah semua bilangan real:<\/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> Sebaliknya, rentang (atau rentang) fungsi kosinus hiperbolik adalah 1 dan semua bilangan lebih besar dari 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> Kosinus hiperbolik adalah fungsi kontinu dan genap.<\/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> Fungsi tersebut memotong sumbu Y di titik 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> Sebaliknya, fungsi tersebut tidak mempunyai titik potong dengan sumbu X.<\/li>\n<\/ul>\n<ul>\n<li> Dua limit hingga tak terhingga (positif dan negatif) dari fungsi kosinus hiperbolik menghasilkan plus tak terhingga.<\/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> Kosinus hiperbolik mengecil hingga x = 0 dan sejak titik tersebut bertambah hingga tak terhingga, sehingga fungsinya mempunyai minimum di 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> Fungsinya cembung di seluruh domainnya, sehingga tidak mempunyai titik belok.<\/li>\n<\/ul>\n<ul>\n<li> Turunan dari fungsi kosinus hiperbolik adalah sinus hiperbolik:<\/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> Integral fungsi kosinus hiperbolik adalah sinus hiperbolik:<\/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> Polinomial Taylor (atau deret Maclaurin) dari fungsi kosinus hiperbolik adalah sebagai berikut:<\/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> Transformasi Laplace dari fungsi kosinus hiperbolik adalah sebagai berikut: <\/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> Hubungan matematis kosinus hiperbolik<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Selanjutnya, kita akan melihat bagaimana kosinus hiperbolik dapat dihitung dari fungsi hiperbolik lainnya, karena semuanya berhubungan secara matematis.<\/p>\n<p> Persamaan dasar menghubungkan kosinus hiperbolik dengan sinus hiperbolik:<\/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>Lihat:<\/strong> <a href=\"https:\/\/mathority.org\/id\/fungsi-sinus-hiperbolik\/\"><span style=\"text-decoration: underline;\">sinus hiperbolik<\/span><\/a><\/p>\n<p> Tiga fungsi hiperbolik utama (sinus hiperbolik, kosinus, dan tangen) dapat dihubungkan dengan persamaan berikut:<\/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> Sebaliknya, kosinus hiperbolik dari penjumlahan (atau pengurangan) dua bilangan berbeda dapat ditentukan dengan rumus berikut:<\/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> Kosinus hiperbolik dua kali suatu bilangan sama dengan jumlah kuadrat kosinus hiperbolik dan sinus hiperbolik bilangan ini:<\/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> Penjumlahan atau pengurangan dua kosinus hiperbolik dapat dihitung dengan menggunakan rumus berikut:<\/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> Terakhir, kuadrat kosinus hiperbolik dapat dihitung dengan rumus berikut:<\/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>Di sini Anda akan menemukan segala sesuatu tentang fungsi kosinus hiperbolik: apa rumusnya, representasi grafisnya, karakteristiknya, hubungan matematisnya dengan fungsi lain, dll. Rumus kosinus hiperbolik Fungsi kosinus hiperbolik adalah salah satu fungsi hiperbolik utama dan diwakili oleh simbol cosh(x) . Kosinus hiperbolik sama dengan jumlah e x ditambah e -x dibagi 2. Oleh karena itu, &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/fungsi-kosinus-hiperbolik\/\"> <span class=\"screen-reader-text\">Fungsi kosinus hiperbolik<\/span> Selengkapnya &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-364","post","type-post","status-publish","format-standard","hentry","category-representasi-fungsi"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Fungsi kosinus hiperbolik - 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\/id\/fungsi-kosinus-hiperbolik\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fungsi kosinus hiperbolik - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di sini Anda akan menemukan segala sesuatu tentang fungsi kosinus hiperbolik: apa rumusnya, representasi grafisnya, karakteristiknya, hubungan matematisnya dengan fungsi lain, dll. Rumus kosinus hiperbolik Fungsi kosinus hiperbolik adalah salah satu fungsi hiperbolik utama dan diwakili oleh simbol cosh(x) . Kosinus hiperbolik sama dengan jumlah e x ditambah e -x dibagi 2. 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