{"id":196,"date":"2019-08-27T20:57:53","date_gmt":"2019-08-27T20:57:53","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=196"},"modified":"2019-12-02T21:22:36","modified_gmt":"2019-12-02T21:22:36","slug":"special-trigonometric-integrals","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/special-trigonometric-integrals\/","title":{"rendered":"Special Trigonometric Integrals"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Special Trigonometric Integrals &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nIn the study of Fourier Series, you will find that every continuous\nfunction $f$ on an interval $[-L,L]$ can be expressed on that interval\nas an infinite series of sines and cosines.  For example, if the\ninterval is $[-\\pi,\\pi]$,\n\\[f(x)=A_0+\\sum^{\\infty}_{k=1} [A_k\\cos (kx)+B_k\\sin\n(kx)]\\]\nwhere the constants are given by integrals involving $f$.\n\n<\/p>\n\n\n\n<p>\nThe theory of Fourier series relies on the fact that the functions\n\\[1,\\quad \\cos x,\\quad \\sin x,\\quad \\cos 2x,\\quad \\sin 2x,\\quad\n\\ldots,\\quad \\cos nx,\\quad \\sin nx,\\quad \\ldots\\]\nform an <b>orthogonal set<\/b>:\n<\/p>\n\n\n\n<center>\n\t<i>The integral of the product of any $2$ of these functions over\n\t$[-\\pi, \\pi]$ is 0.<\/i>\n<\/center>\n\n\n\n<p>\nHere, we will verify this fact.\n\n<\/p>\n\n\n\n<p>\nWe will use the following trigonometric identities:\n\\begin{eqnarray*}\n\t\\sin A\\sin B =\\frac{1}{2}[\\cos (A-B)-\\cos (A+B)] \\phantom{.}\\\\\n\t\\cos A\\cos B =\\frac{1}{2}[\\cos (A-B)+\\cos (A+B)] \\phantom{.}\\\\\n\t\\sin A\\cos B =\\frac{1}{2}[\\sin (A-B)+\\sin (A+B)].\n\\end{eqnarray*}\nWe have six general integrals to evaluate to prove the orthogonality\nof the set $\\{1, ~\\cos x, ~\\sin x, \\ldots\\}$.  In each of the following,\nwe assume $m$ and $n$ are distinct positive integers.\n\n<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li> $\\quad\\displaystyle\\int^\\pi_{-\\pi} 1\\cdot \\cos (nx)\\,\n\t\tdx=\\left.\\frac{1}{n}\\sin (nx)\\right|^{\\pi}_{-\\pi}=0.$\n\t<br><br>\n\t<\/li><li> $\\quad\\displaystyle\\int^\\pi_{-\\pi} 1\\cdot \\sin (nx)\\,\n\t\tdx=\\left.-\\frac{1}{n}\\cos (nx)\\right|^{\\pi}_{-\\pi}=0.$\n\t<br><br>\n\t<\/li><li> $\\quad\\displaystyle\\int^\\pi_{-\\pi} \\sin (nx)\\cos (nx)\\,\n\t\tdx=\\left.\\frac{\\sin^2 (nx)}{2n}\\right|^{\\pi}_{-\\pi}=0.$\n\t<br><br>\n\t<\/li><li> &nbsp;\n\t\t\\begin{eqnarray*}\n\t\t\t\\int^\\pi_{-\\pi} \\sin (mx)\\sin\n\t\t\t(nx)\\,dx&amp;=&amp;\\int^{\\pi}_{-\\pi}\\frac{1}{2}[\\cos (m-n)x-\\cos (m+n)x]\\,\n\t\t\tdx\\\\\n\t\t\t&amp;=&amp;\\left.\\left(\\frac{\\sin[(m-n)x]}{2(m-n)}-\n\t\t\t\\frac{\\sin[(m+n)x]}{2(m+n)}\\right)\\right|^{\\pi}_{-\\pi}\\\\\n\t\t\t&amp;=&amp;0.\n\t\t\\end{eqnarray*}\n\t<br><br>\n\t<\/li><li> &nbsp;\n\t\t\\begin{eqnarray*}\n\t\t\t\\int^\\pi_{-\\pi} \\cos (mx)\\cos\n\t\t\t(nx)\\,dx&amp;=&amp;\\int^{\\pi}_{-\\pi}\\frac{1}{2}[\\cos (m-n)x+\\cos (m+n)x]\\,\n\t\t\tdx\\\\\n\t\t\t&amp;=&amp;\\left.\\left(\\frac{\\sin[(m-n)x]}{2(m-n)}+\n\t\t\t\\frac{\\sin[(m+n)x]}{2(m+n)}\\right)\\right|^{\\pi}_{-\\pi}\\\\\n\t\t\t&amp;=&amp;0.\n\t\t\\end{eqnarray*}\n\t<br><br>\n\t<\/li><li> &nbsp;\n\t\t\\begin{eqnarray*}\n\t\t\t\\int^\\pi_{-\\pi} \\sin (mx)\\cos\n\t\t\t(nx)\\,dx&amp;=&amp;\\int^{\\pi}_{-\\pi}\\frac{1}{2}[\\sin (m-n)x+\\sin (m+n)x]\\,\n\t\t\tdx\\\\\n\t\t\t&amp;=&amp;\\left.\\left(-\\frac{\\cos[(m-n)x]}{2(m-n)}-\n\t\t\t\\frac{\\cos[(m+n)x]}{2(m+n)}\\right)\\right|^{\\pi}_{-\\pi}\\\\\n\t\t\t&amp;=&amp;0.\n\t\t\\end{eqnarray*}\n<\/li><\/ol>\n\n\n\n<p>\nWe have now shown that $\\{1, ~\\cos x, ~\\sin x, ~\\cos 2x, ~\\sin 2x, \\ldots\\}$\nis indeed an orthogonal set of functions!\n\n<\/p>\n\n\n\n<p>\nIn the following Exploration, graph functions $\\sin(mx)\\sin(nx)$, \n$\\sin(mx)\\cos(nx)$, and $\\cos(mx)\\cos(nx)$ for various values of m and n and \nobserve the interesting curves that result.\n\n<\/p>\n\n\n\n<p> The theory of Fourier series relies on the fact that the functions <br> $1, ~\\cos x, ~\\sin x, ~\\cos 2x, ~\\sin 2x, ~\\ldots, ~\\cos nx, ~\\sin nx, \\ldots$ form an orthogonal set.<\/p>\n\n\n\n<p>\nThe integral of the product of any $2$ of these functions over $[-\\pi,\\pi]$ is \n$0$.\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ0510\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Special Trigonometric Integrals &#8211; HMC Calculus Tutorial In the study of Fourier Series, you will find that every continuous function $f$ on an interval $[-L,L]$ can be expressed on that interval as an infinite series of sines and cosines. For example, if the interval is $[-\\pi,\\pi]$, \\[f(x)=A_0+\\sum^{\\infty}_{k=1} [A_k\\cos (kx)+B_k\\sin (kx)]\\] where the constants are given&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-196","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/196","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=196"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/196\/revisions"}],"predecessor-version":[{"id":1098,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/196\/revisions\/1098"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=196"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}