{"id":156,"date":"2019-08-15T17:38:14","date_gmt":"2019-08-15T17:38:14","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=156"},"modified":"2020-06-17T18:49:49","modified_gmt":"2020-06-17T18:49:49","slug":"review-of-trig-log-exp","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/","title":{"rendered":"Review of Trig, Log, Exp"},"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<p>\n<!------------------------>\n\nIn this tutorial, we review trigonometric, logarithmic, and\nexponential functions with a focus on those properties which will be useful\nin future math and science applications.\n\n<\/p>\n\n\n\n<p>\n\nGeometrically, there are two ways to describe trigonometric\nfunctions: \n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Polar Angle<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"347\" height=\"313\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2020\/01\/unit_circle.gif?resize=347%2C313&#038;ssl=1\" alt=\"\" class=\"wp-image-1157\"\/><\/figure><\/div>\n\n\n\n<p>\n\n\n$\\begin{array}{l}\n\tx=\\cos\\theta\\\\\n\ty=\\sin\\theta\\\\\n\t~\\\\\n\t{\\small\\textrm{Measure }} \\theta {\\small\\textrm{ in radians:}}\\\\\n\t\\theta =\\frac{{\\small\\textrm{arc length}}}{{\\small\\textrm{radius}}}\\\\\n\t~\\\\\n\t{\\small\\textrm{For example,}}\\quad 180^{\\circ}=\\displaystyle\\frac{\\pi r}{r}=\\pi\n\t{\\small\\textrm{ radians}}\\\\\n\t~\\\\\n\t{\\small\\textrm{Radians}}=\\displaystyle\\frac{{\\small\\textrm{degrees}}}{180}\\cdot \\pi\n\\end{array}$\n\n<br><br><br><br><br>\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Right Angle<\/h4>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"298\" height=\"185\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2020\/01\/right_triangle.gif?resize=298%2C185&#038;ssl=1\" alt=\"\" class=\"wp-image-1156\"\/><\/figure><\/div>\n\n\n\n<p>\n\n\n$\\begin{array}{l}\n\t\\sin\\theta &amp;=&amp; \\displaystyle \\frac{{\\small\\textrm{opposite}}}{{\\small\\textrm{hypotenuse}}} &amp;=&amp; \\frac{y}{r}\\\\~\\\\\n\t\\cos\\theta &amp;=&amp; \\displaystyle \\frac{{\\small\\textrm{adjacent}}}{{\\small\\textrm{hypotenuse}}} &amp;=&amp; \\frac{x}{r}\\\\~\\\\\n\t\\tan\\theta &amp;=&amp; \\displaystyle \\frac{{\\small\\textrm{opposite}}}{{\\small\\textrm{adjacent}}} &amp;=&amp; \\frac{y}{x}\\\\~\\\\\n\t\\csc\\theta &amp;=&amp; \\displaystyle \\frac{1}{\\sin\\theta} &amp;=&amp; \\frac{r}{y}\\\\~\\\\\n\t\\sec\\theta &amp;=&amp; \\displaystyle \\frac{1}{\\cos\\theta} &amp;=&amp; \\frac{r}{x}\\\\~\\\\\n\t\\cot\\theta &amp;=&amp; \\displaystyle \\frac{1}{\\tan\\theta} &amp;=&amp; \\frac{x}{y}\n\\end{array}$\n\t\n<br><br>\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Evaluating Trigonometric Functions<\/h4>\n\n\n\n<center>\n<table align=center width=100% rules=groups>\n\t\t<colgroup><\/colgroup>\n\t\t<colgroup colspan=5><\/colgroup>\n\n\t\t<tbody>\n\t\t\t<tr>\n\t\t\t\t<td align=center> \n\t\t\t\t<td align=center> $0 {\\small\\textrm{ rad}} $\n\t\t\t\t<td align=center> $\\pi\/6 {\\small\\textrm{ rad}} $\n\t\t\t\t<td align=center> $\\pi\/4 {\\small\\textrm{ rad}} $\n\t\t\t\t<td align=center> $\\pi\/3 {\\small\\textrm{ rad}}  $\n\t\t\t\t<td align=center> $\\pi\/2 {\\small\\textrm{ rad}}$\n\t\t\t<tr>\n\t\t\t\t<td align=center> \n\t\t\t\t<td align=center> $0^{\\circ} $\n\t\t\t\t<td align=center> $30^{\\circ} $\n\t\t\t\t<td align=center> $45^{\\circ} $\n\t\t\t\t<td align=center> $60^{\\circ} $\n\t\t\t\t<td align=center> $90^{\\circ}$\n\t\t<\/tbody>\n\n\t\t<tbody>\n\t\t\t<tr>\n\t\t\t\t<td align=center> $\\sin\\theta $\n\t\t\t\t<td align=center> $0 $\n\t\t\t\t<td align=center> $1\/2 $\n\t\t\t\t<td align=center> $\\sqrt{2}\/2 $\n\t\t\t\t<td align=center> $\\sqrt{3}\/2 $\n\t\t\t\t<td align=center> $1$\n\t\t\t<tr>\n\t\t\t\t<td align=center> $\\cos\\theta $\n\t\t\t\t<td align=center> $1 $\n\t\t\t\t<td align=center> $\\sqrt{3}\/2 $\n\t\t\t\t<td align=center> $\\sqrt{2}\/2 $\n\t\t\t\t<td align=center> $1\/2 $\n\t\t\t\t<td align=center> $0$\n\t\t\t<tr>\n\t\t\t\t<td align=center> $\\tan\\theta $\n\t\t\t\t<td align=center> $0 $\n\t\t\t\t<td align=center> $\\sqrt{3}\/3 $\n\t\t\t\t<td align=center> $1 $\n\t\t\t\t<td align=center> $\\sqrt{3} $\n\t\t\t\t<td align=center> ${\\small\\textrm{undefined}}$\n\t\t<\/tbody>\n\t<\/table>\n<\/center>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"171\" height=\"180\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2020\/01\/45triangle.gif?resize=171%2C180&#038;ssl=1\" alt=\"\" class=\"wp-image-1154\"\/><\/figure><\/div>\n\n\n\n$\\begin{array}{lcr}\n\t\t\t\t\\sin(-\\theta) &amp; = &amp; -\\sin\\theta \\\\\n\t\t\t\t\\cos(-\\theta)&amp; = &amp; \\cos\\theta \\\\\n\t\t\t\t~\\\\\n\t\t\t\t\\cos(\\theta+\\pi) &amp; = &amp; -\\cos\\theta \\\\\n\t\t\t\t\\sin(\\theta+\\pi) &amp; = &amp; -\\sin\\theta \n\t\t\t\\end{array}$\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"296\" height=\"181\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2020\/01\/3060triangle.gif?resize=296%2C181&#038;ssl=1\" alt=\"\" class=\"wp-image-1155\"\/><\/figure><\/div>\n\n\n\n$\\begin{array}{lcr}\n\t\t\t\t\\sin(\\theta +\\pi\/2) &amp; = &amp; \\cos\\theta\\\\ \n\t\t\t\t\\cos(\\theta +\\pi\/2) &amp; = &amp; -\\sin\\theta\\\\\n\t\t\t\t~\\\\\n\t\t\t\t\\cos(\\theta +2\\pi) &amp; = &amp; \\cos\\theta\\\\\n\t\t\t\t\\sin(\\theta +2\\pi) &amp; = &amp; \\sin\\theta\n\t\t\t\\end{array}$\n\n\n\n<p><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Trigonometric Identities<\/h4>\n\n\n\n<p>\nWe list here some of the most commonly used identities:\n\n<\/p>\n\n\n\n<p>\n\t\t$\\begin{array}{lcr}\n\t\t\t\\textbf{1.  }\\cos^2\\theta+\\sin^2\\theta=1 \\\\\n\t\t\t\\textbf{2.  }\\cos^2\\theta =\\displaystyle\\frac{1}{2}[1+\\cos(2\\theta)] \\\\\n\t\t\t\\textbf{3.  } \\sin^2\\theta=\\displaystyle\\frac{1}{2}[1-\\cos(2\\theta)]\\\\\n\t\t\t\\textbf{4.  } \\sin(2\\theta)=2\\sin\\theta\\cos\\theta \\\\ \n\t\t\t\\textbf{5.  }\\cos(2\\theta)=\\cos^2\\theta-\\sin^2\\theta \n\t\t\\end{array}$\n\t\n<\/p>\n\n\n\n<p>\n\t\t$\\begin{array}{lcr}\n\t\t\t\\textbf{6.  } \\sin(\\alpha+\\beta)=\\sin\\alpha\\cos\\beta+\\cos\\alpha\\sin\\beta\\\\\n\t\t\t\\textbf{7.  } \\cos(\\alpha +\\beta)=\\cos\\alpha\\cos\\beta-\\sin\\alpha\\sin\\beta\\\\\n\t\t\t~\\\\\n\t\t\t\\textbf{8.  } C_1\\cos(\\omega x)+C_2\\sin(\\omega x)=A\\sin(\\omega x+\\phi)\\\\\n\t\t\t\\qquad{\\small\\textrm{where }} A=\\sqrt{C_1^2+C_2^2},\\quad \\phi=\\arctan (C_1\/C_2)\n\t\t\\end{array}$\n<br><\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Logarithmic and Exponential Functions<\/h4>\n\n\n\n<p>\nLogarithmic and exponential functions are inverses of each other:\n\\begin{eqnarray*}\n\ty=\\log_b x &amp; \\quad{\\small\\textrm{if and only if}} &amp; x=b^y\\\\\n\ty=\\ln x &amp; {\\small\\textrm{ if and only if }} &amp; x=e^y.\n\\end{eqnarray*}\nIn words, $\\displaystyle \\log_b x$ is the exponent you put on base $b$\nto get $x$.  Thus, \n\\[log_b b^x=x \\qquad {\\small\\textrm{and}} \\qquad b^{\\log_b x}=x.\\]\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\n<b>More Properties of Logarithmic and Exponential Functions<\/b>\n\n<\/h4>\n\n\n\n<p>\nNotice the relationship between each pair of identities:\n\\[\\begin{array}{ccc@{\\qquad}ccc}\n\t\\log_b 1=0 &amp; \\longleftrightarrow &amp; b^0=1 &amp; \\log_b ac=\\log_b a+\\log_b c\n\t&amp; \\longleftrightarrow &amp; b^mb^n=b^{m+n}\\\\\n\t\\log_b b=1 &amp; \\longleftrightarrow &amp; b^1=b &amp; \\log_b\n\t\\displaystyle\\frac{a}{c}=\\log_b \n\ta-\\log_b c &amp; \\longleftrightarrow &amp; \\displaystyle\\frac{b^m}{b^n}=b^{m-n}\\\\\n\t\\log_b \\displaystyle\\frac{1}{c}=-\\log_b c &amp; \\longleftrightarrow &amp;\n\tb^{-m}=\\displaystyle\\frac{1}{b^m} &amp; \n\t\\log_b a^r=r\\log_b a &amp; \\longleftrightarrow &amp; (b^m)^n=b^{mn} .\n\\end{array}\\]\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\n<b>Graphs of Logarithmic and Exponential Functions<\/b>\n\n<\/h4>\n\n\n\n<ul class=\"wp-block-gallery columns-2 is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\"><li class=\"blocks-gallery-item\"><figure><a href=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/sinx-2\/#main\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"290\" height=\"139\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2020\/06\/sinx.gif?resize=290%2C139&#038;ssl=1\" alt=\"\" data-id=\"1201\" data-link=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/sinx-2\/#main\" class=\"wp-image-1201\"\/><\/a><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><a href=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/cosx-2\/#main\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"339\" height=\"186\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2020\/06\/cosx.gif?resize=339%2C186&#038;ssl=1\" alt=\"\" data-id=\"1202\" data-link=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/cosx-2\/#main\" class=\"wp-image-1202\"\/><\/a><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><a href=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/tanx-2\/#main\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"328\" height=\"263\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2020\/06\/tanx.gif?resize=328%2C263&#038;ssl=1\" alt=\"\" data-id=\"1203\" data-link=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/tanx-2\/#main\" class=\"wp-image-1203\"\/><\/a><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><a href=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/cotx-2\/#main\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"315\" height=\"321\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2020\/06\/cotx.gif?resize=315%2C321&#038;ssl=1\" alt=\"\" data-id=\"1204\" data-link=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/cotx-2\/#main\" class=\"wp-image-1204\"\/><\/a><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><a href=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/secx-2\/#main\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"315\" height=\"321\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2020\/06\/secx.gif?resize=315%2C321&#038;ssl=1\" alt=\"\" data-id=\"1205\" data-link=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/secx-2\/#main\" class=\"wp-image-1205\"\/><\/a><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><a href=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/cscx-2\/#main\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"315\" height=\"321\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2020\/06\/cscx.gif?resize=315%2C321&#038;ssl=1\" alt=\"\" data-id=\"1199\" data-link=\"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/review-of-trig-log-exp\/cscx-2\/#main\" class=\"wp-image-1199\"\/><\/a><\/figure><\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">\n<b>Limits of Logarithmic and Exponential Functions<\/b>\n\n<\/h4>\n\n\n\n<ol class=\"wp-block-list\"><li> $\\displaystyle \\lim_{x\\to\\infty} \\frac{\\ln x}{x}=0\\quad$ $\\ln x$ grows more slowly than $x$. <br><br> <\/li><li> $\\displaystyle \\lim_{x\\to\\infty} \\frac{e^x}{x^n}=\\infty$ for all positive integers $n\\quad$, $\\displaystyle e^x$ grows faster than $x^n$. <br><br> <\/li><li> For $|x|\\ll 1$, $\\displaystyle\\lim_{n\\to\\infty} \\left(1+\\frac{x}{n}\\right)^n=e^x$.  <\/li><\/ol>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ3210\/\">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>In this tutorial, we review trigonometric, logarithmic, and exponential functions with a focus on those properties which will be useful in future math and science applications. Geometrically, there are two ways to describe trigonometric functions: Polar Angle $\\begin{array}{l} x=\\cos\\theta\\\\ y=\\sin\\theta\\\\ ~\\\\ {\\small\\textrm{Measure }} \\theta {\\small\\textrm{ in radians:}}\\\\ \\theta =\\frac{{\\small\\textrm{arc length}}}{{\\small\\textrm{radius}}}\\\\ ~\\\\ {\\small\\textrm{For example,}}\\quad 180^{\\circ}=\\displaystyle\\frac{\\pi r}{r}=\\pi&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":55,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-156","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/156","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=156"}],"version-history":[{"count":26,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/156\/revisions"}],"predecessor-version":[{"id":1207,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/156\/revisions\/1207"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/55"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=156"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=156"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}