{"id":193,"date":"2025-02-13T22:44:52","date_gmt":"2025-02-13T22:44:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-other-trigonometric-functions\/"},"modified":"2025-10-13T20:47:14","modified_gmt":"2025-10-13T20:47:14","slug":"graphs-of-the-other-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-other-trigonometric-functions\/","title":{"raw":"Graphs of the Other Trigonometric Functions: Learn It 1","rendered":"Graphs of the Other Trigonometric Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Graph transformations of y=tan x and y=cot x.<\/li>\r\n \t<li style=\"font-weight: 400;\">Determine a function formula from a tangent or cotangent graph.<\/li>\r\n \t<li style=\"font-weight: 400;\">Graph transformations of y=sec x and y=csc x.<\/li>\r\n \t<li style=\"font-weight: 400;\">Determine a function formula from a secant or cosecant graph.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Analyzing the Graph of y = tan x and Its Variations<\/h2>\r\nWe will begin with the graph of the <strong>tangent<\/strong> function, plotting points as we did for the sine and cosine functions. Recall that\r\n<div>\r\n<div style=\"text-align: center;\">[latex]\\tan x=\\frac{\\sin x}{\\cos x}[\/latex]<\/div>\r\n<\/div>\r\nThe <strong>period<\/strong> of the tangent function is <em>\u03c0<\/em> because the graph repeats itself on intervals of <em>k\u03c0<\/em> where <em>k<\/em> is a constant. If we graph the tangent function on [latex]\u2212\\dfrac{\\pi}{2}\\text{ to }\\dfrac{\\pi}{2}[\/latex], we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.\r\n\r\nWe can determine whether tangent is an odd or even function by using the definition of tangent.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan(\u2212x)&amp;=\\frac{\\sin(\u2212x)}{\\cos(\u2212x)} &amp;&amp; \\text{Definition of tangent.} \\\\ &amp;=\\frac{\u2212\\sin x}{\\cos x} &amp;&amp; \\text{Sine is an odd function, cosine is even.} \\\\ &amp;=\u2212\\frac{\\sin x}{\\cos x} &amp;&amp; \\text{The quotient of an odd and an even function is odd.} \\\\ &amp;=\u2212\\tan x &amp;&amp; \\text{Definition of tangent.} \\end{align}[\/latex]<\/p>\r\nTherefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in the table below.\r\n<table id=\"Table_06_02_00\" style=\"width: 1035px;\" summary=\"Two rows and 10 columns. First row is labeled x and second row is labeled tangent of x. The table has ordered pairs of these column values: (-pi\/2,undefined), (-pi\/3, negative square root of 3), (-pi\/4, -1), (-pi\/6, negative square root of 3 over 3), (0, 0), (pi\/6, square root of 3 over 3), (pi\/4, 1), (pi\/3, square root of 3), (pi\/2, undefined).\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 30px;\"><em><strong> x <\/strong><\/em><\/td>\r\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">0<\/td>\r\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 30px;\"><strong> tan (<em>x<\/em>) <\/strong><\/td>\r\n<td style=\"width: 80px;\">undefined<\/td>\r\n<td style=\"width: 80px;\">[latex]\u2212\\sqrt{3}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">\u20131<\/td>\r\n<td style=\"width: 80px;\">[latex]\u2212\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">0<\/td>\r\n<td style=\"width: 80px;\">[latex]\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">1<\/td>\r\n<td style=\"width: 80px;\">[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td style=\"width: 80px;\">undefined<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when [latex]\\frac{\\pi}{3}&lt;x&lt;\\frac{\\pi}{2}[\/latex], we can use a table to look for a trend. Because [latex]\\frac{\\pi}{3}\\approx 1.05[\/latex] and [latex]\\frac{\\pi}{2}\\approx 1.57[\/latex], we will evaluate x at radian measures 1.05 &lt; <em>x<\/em> &lt; 1.57 as shown in the table below.\r\n<table id=\"Table_06_02_01\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (1.3, 3.6), (1.5, 14.1), (1.55, 48.1), (1.56, 92.6).\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong> x <\/strong><\/em><\/td>\r\n<td>1.3<\/td>\r\n<td>1.5<\/td>\r\n<td>1.55<\/td>\r\n<td>1.56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong> tan <em>x <\/em><\/strong><\/td>\r\n<td>3.6<\/td>\r\n<td>14.1<\/td>\r\n<td>48.1<\/td>\r\n<td>92.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs <em>x<\/em> approaches [latex]\\frac{\\pi}{2}[\/latex], the outputs of the function get larger and larger. Because [latex]y=\\tan x[\/latex] is an odd function, we see the corresponding table of negative values in the table below.\r\n<table id=\"Table_06_02_02\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (-1.3, -3.6), (-1.5, -14.1), (-1.55, -48.1), (-1.56, -92.6).\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong> x <\/strong><\/em><\/td>\r\n<td>\u22121.3<\/td>\r\n<td>\u22121.5<\/td>\r\n<td>\u22121.55<\/td>\r\n<td>\u22121.56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong> tan <em>x <\/em><\/strong><\/td>\r\n<td>\u22123.6<\/td>\r\n<td>\u221214.1<\/td>\r\n<td>\u221248.1<\/td>\r\n<td>\u221292.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can see that, as <em>x<\/em> approaches [latex]\u2212\\dfrac{\\pi}{2}[\/latex], the outputs get smaller and smaller. Remember that there are some values of <em>x<\/em> for which cos <em>x<\/em> = 0. For example, [latex]\\cos\\left(\\frac{\\pi}{2}\\right)=0[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)=0[\/latex]. At these values, the <strong>tangent function<\/strong> is undefined, so the graph of [latex]y=\\tan x[\/latex] has discontinuities at [latex]x=\\frac{\\pi}{2}[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex]. At these values, the graph of the tangent has vertical asymptotes. The tangent is positive from 0 to [latex]\\frac{\\pi}{2}[\/latex] and from <em>\u03c0<\/em> to [latex]\\frac{3\\pi}{2}[\/latex], corresponding to quadrants I and III of the unit circle.\r\n<figure id=\"Figure_06_02_001\" class=\"small ui-has-child-figcaption\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163804\/CNX_Precalc_Figure_06_02_001.jpg\" alt=\"A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.\" width=\"487\" height=\"316\" \/> Graph of the tangent function[\/caption]<\/figure>\r\nAs with the sine and cosine functions, the <strong>tangent<\/strong> function can be described by a general equation.\r\n<div>\r\n<div style=\"text-align: center;\">[latex]y=A\\tan(Bx)[\/latex]<\/div>\r\n<\/div>\r\nWe can identify horizontal and vertical stretches and compressions using values of A and B. The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.\r\n\r\nBecause there are no maximum or minimum values of a tangent function, the term <em>amplitude<\/em> cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase <em>stretching\/compressing factor<\/em> when referring to the constant A.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">features of the graph of <\/span><em style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">y<\/em><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\"> = <\/span><em style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">A<\/em><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">tan(<\/span><em style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">Bx<\/em><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">)<\/span><\/h3>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>| (tangent does not have an amplitude)<\/li>\r\n \t<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is all real numbers <em>x<\/em>, where [latex]x\\ne \\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|} k[\/latex] such that <em>k<\/em> is an integer.<\/li>\r\n \t<li>The range is [latex]\\left(-\\infty,\\infty\\right)[\/latex].<\/li>\r\n \t<li>The asymptotes occur at [latex]x=\\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>[latex]y = A \\tan (Bx)[\/latex] is an odd function.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Graphing One Period of a Stretched or Compressed Tangent Function<\/h2>\r\nWe can use what we know about the properties of the <strong>tangent function<\/strong> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form [latex]f(x)=A\\tan(Bx)[\/latex]. We focus on a single <strong>period<\/strong> of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function\u2019s domain if we wish. Our limited domain is then the interval [latex](\u2212\\frac{P}{2}, \\frac{P}{2})[\/latex] and the graph has vertical asymptotes at [latex]\\pm \\frac{P}{2}[\/latex] where [latex]P=\\frac{\\pi}{B}[\/latex]. On [latex](\u2212\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})[\/latex], the graph will come up from the left asymptote at [latex]x=\u2212\\dfrac{\\pi}{2}[\/latex], cross through the origin, and continue to increase as it approaches the right asymptote at [latex]x=\\frac{\\pi}{2}[\/latex]. To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use\r\n<div>\r\n<div style=\"text-align: center;\">[latex]f\\left(\\frac{P}{4}\\right)=A \\tan\\left(B\\frac{P}{4}\\right)=A\\tan\\left(B\\frac{\\pi}{4B}\\right)=A[\/latex]<\/div>\r\n<\/div>\r\nbecause \u00a0[latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex].\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the function [latex]f(x)=A\\tan(Bx)[\/latex], graph one period.<\/strong>\r\n<ol>\r\n \t<li>Identify the stretching factor, |A|.<\/li>\r\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Draw vertical asymptotes at \u00a0[latex]x=\u2212\\dfrac{P}{2}[\/latex] and [latex]x=\\frac{P}{2}[\/latex].<\/li>\r\n \t<li>For <em>A<\/em> &gt; 0 , the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for <em>A<\/em> &lt; 0 ).<\/li>\r\n \t<li>Plot reference points at [latex]\\left(\\frac{P}{4},A\\right)[\/latex]\u00a0(0, 0), and ([latex]\u2212\\dfrac{P}{4}[\/latex],\u2212 A), and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Sketch a graph of one period of the function [latex]y=0.5\\tan\\left(\\frac{\\pi}{2}x\\right)[\/latex].[reveal-answer q=\"302986\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"302986\"]First, we identify <em>A<\/em> and B.<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163807\/CNX_Precalc_Figure_06_02_002.jpg\" alt=\"An illustration of equations showing that A is the coefficient of tangent and B is the coefficient of x, which is within the tangent function.\" width=\"487\" height=\"113\" \/>\r\n\r\nBecause [latex]A=0.5[\/latex] and [latex]B=\\frac{\\pi}{2}[\/latex], we can find the <strong>stretching\/compressing factor<\/strong> and period. The period is [latex]\\frac{\\pi}{\\frac{\\pi}{2}}=2[\/latex], so the asymptotes are at [latex]x=\\pm 1[\/latex]. At a quarter period from the origin, we have\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f(0.5)&amp;=0.5\\tan\\left(\\frac{0.5\\pi}{2}\\right)\\\\ &amp;=0.5\\tan(\\frac{\\pi}{4})\\\\ &amp;=0.5 \\end{align}[\/latex]<\/p>\r\nThis means the curve must pass through the points(0.5,0.5),(0,0),and(\u22120.5,\u22120.5).The only inflection point is at the origin. Below is the graph of one period of the function.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163810\/CNX_Precalc_Figure_06_02_003.jpg\" alt=\"A graph of one period of a modified tangent function, with asymptotes at x=-1 and x=1.\" width=\"487\" height=\"258\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nSketch a graph of [latex]f(x)=3\\tan\\left(\\frac{\\pi}{6}x\\right)[\/latex].\r\n\r\n[reveal-answer q=\"547078\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"547078\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163917\/CNX_Precalc_Figure_06_02_004.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]174880[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Graph transformations of y=tan x and y=cot x.<\/li>\n<li style=\"font-weight: 400;\">Determine a function formula from a tangent or cotangent graph.<\/li>\n<li style=\"font-weight: 400;\">Graph transformations of y=sec x and y=csc x.<\/li>\n<li style=\"font-weight: 400;\">Determine a function formula from a secant or cosecant graph.<\/li>\n<\/ul>\n<\/section>\n<h2>Analyzing the Graph of y = tan x and Its Variations<\/h2>\n<p>We will begin with the graph of the <strong>tangent<\/strong> function, plotting points as we did for the sine and cosine functions. Recall that<\/p>\n<div>\n<div style=\"text-align: center;\">[latex]\\tan x=\\frac{\\sin x}{\\cos x}[\/latex]<\/div>\n<\/div>\n<p>The <strong>period<\/strong> of the tangent function is <em>\u03c0<\/em> because the graph repeats itself on intervals of <em>k\u03c0<\/em> where <em>k<\/em> is a constant. If we graph the tangent function on [latex]\u2212\\dfrac{\\pi}{2}\\text{ to }\\dfrac{\\pi}{2}[\/latex], we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.<\/p>\n<p>We can determine whether tangent is an odd or even function by using the definition of tangent.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\tan(\u2212x)&=\\frac{\\sin(\u2212x)}{\\cos(\u2212x)} && \\text{Definition of tangent.} \\\\ &=\\frac{\u2212\\sin x}{\\cos x} && \\text{Sine is an odd function, cosine is even.} \\\\ &=\u2212\\frac{\\sin x}{\\cos x} && \\text{The quotient of an odd and an even function is odd.} \\\\ &=\u2212\\tan x && \\text{Definition of tangent.} \\end{align}[\/latex]<\/p>\n<p>Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in the table below.<\/p>\n<table id=\"Table_06_02_00\" style=\"width: 1035px;\" summary=\"Two rows and 10 columns. First row is labeled x and second row is labeled tangent of x. The table has ordered pairs of these column values: (-pi\/2,undefined), (-pi\/3, negative square root of 3), (-pi\/4, -1), (-pi\/6, negative square root of 3 over 3), (0, 0), (pi\/6, square root of 3 over 3), (pi\/4, 1), (pi\/3, square root of 3), (pi\/2, undefined).\">\n<tbody>\n<tr>\n<td style=\"width: 30px;\"><em><strong> x <\/strong><\/em><\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">0<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30px;\"><strong> tan (<em>x<\/em>) <\/strong><\/td>\n<td style=\"width: 80px;\">undefined<\/td>\n<td style=\"width: 80px;\">[latex]\u2212\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">\u20131<\/td>\n<td style=\"width: 80px;\">[latex]\u2212\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">0<\/td>\n<td style=\"width: 80px;\">[latex]\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">1<\/td>\n<td style=\"width: 80px;\">[latex]\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px;\">undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when [latex]\\frac{\\pi}{3}<x<\\frac{\\pi}{2}[\/latex], we can use a table to look for a trend. Because [latex]\\frac{\\pi}{3}\\approx 1.05[\/latex] and [latex]\\frac{\\pi}{2}\\approx 1.57[\/latex], we will evaluate x at radian measures 1.05 &lt; <em>x<\/em> &lt; 1.57 as shown in the table below.<\/p>\n<table id=\"Table_06_02_01\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (1.3, 3.6), (1.5, 14.1), (1.55, 48.1), (1.56, 92.6).\">\n<tbody>\n<tr>\n<td><em><strong> x <\/strong><\/em><\/td>\n<td>1.3<\/td>\n<td>1.5<\/td>\n<td>1.55<\/td>\n<td>1.56<\/td>\n<\/tr>\n<tr>\n<td><strong> tan <em>x <\/em><\/strong><\/td>\n<td>3.6<\/td>\n<td>14.1<\/td>\n<td>48.1<\/td>\n<td>92.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As <em>x<\/em> approaches [latex]\\frac{\\pi}{2}[\/latex], the outputs of the function get larger and larger. Because [latex]y=\\tan x[\/latex] is an odd function, we see the corresponding table of negative values in the table below.<\/p>\n<table id=\"Table_06_02_02\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (-1.3, -3.6), (-1.5, -14.1), (-1.55, -48.1), (-1.56, -92.6).\">\n<tbody>\n<tr>\n<td><em><strong> x <\/strong><\/em><\/td>\n<td>\u22121.3<\/td>\n<td>\u22121.5<\/td>\n<td>\u22121.55<\/td>\n<td>\u22121.56<\/td>\n<\/tr>\n<tr>\n<td><strong> tan <em>x <\/em><\/strong><\/td>\n<td>\u22123.6<\/td>\n<td>\u221214.1<\/td>\n<td>\u221248.1<\/td>\n<td>\u221292.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can see that, as <em>x<\/em> approaches [latex]\u2212\\dfrac{\\pi}{2}[\/latex], the outputs get smaller and smaller. Remember that there are some values of <em>x<\/em> for which cos <em>x<\/em> = 0. For example, [latex]\\cos\\left(\\frac{\\pi}{2}\\right)=0[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)=0[\/latex]. At these values, the <strong>tangent function<\/strong> is undefined, so the graph of [latex]y=\\tan x[\/latex] has discontinuities at [latex]x=\\frac{\\pi}{2}[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex]. At these values, the graph of the tangent has vertical asymptotes. The tangent is positive from 0 to [latex]\\frac{\\pi}{2}[\/latex] and from <em>\u03c0<\/em> to [latex]\\frac{3\\pi}{2}[\/latex], corresponding to quadrants I and III of the unit circle.<\/p>\n<figure id=\"Figure_06_02_001\" class=\"small ui-has-child-figcaption\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163804\/CNX_Precalc_Figure_06_02_001.jpg\" alt=\"A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.\" width=\"487\" height=\"316\" \/><figcaption class=\"wp-caption-text\">Graph of the tangent function<\/figcaption><\/figure>\n<\/figure>\n<p>As with the sine and cosine functions, the <strong>tangent<\/strong> function can be described by a general equation.<\/p>\n<div>\n<div style=\"text-align: center;\">[latex]y=A\\tan(Bx)[\/latex]<\/div>\n<\/div>\n<p>We can identify horizontal and vertical stretches and compressions using values of A and B. The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.<\/p>\n<p>Because there are no maximum or minimum values of a tangent function, the term <em>amplitude<\/em> cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase <em>stretching\/compressing factor<\/em> when referring to the constant A.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">features of the graph of <\/span><em style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">y<\/em><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\"> = <\/span><em style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">A<\/em><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">tan(<\/span><em style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">Bx<\/em><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 26px; font-weight: 600;\">)<\/span><\/h3>\n<ul>\n<li>The stretching factor is |<em>A<\/em>| (tangent does not have an amplitude)<\/li>\n<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is all real numbers <em>x<\/em>, where [latex]x\\ne \\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|} k[\/latex] such that <em>k<\/em> is an integer.<\/li>\n<li>The range is [latex]\\left(-\\infty,\\infty\\right)[\/latex].<\/li>\n<li>The asymptotes occur at [latex]x=\\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>[latex]y = A \\tan (Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/section>\n<h2>Graphing One Period of a Stretched or Compressed Tangent Function<\/h2>\n<p>We can use what we know about the properties of the <strong>tangent function<\/strong> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form [latex]f(x)=A\\tan(Bx)[\/latex]. We focus on a single <strong>period<\/strong> of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function\u2019s domain if we wish. Our limited domain is then the interval [latex](\u2212\\frac{P}{2}, \\frac{P}{2})[\/latex] and the graph has vertical asymptotes at [latex]\\pm \\frac{P}{2}[\/latex] where [latex]P=\\frac{\\pi}{B}[\/latex]. On [latex](\u2212\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})[\/latex], the graph will come up from the left asymptote at [latex]x=\u2212\\dfrac{\\pi}{2}[\/latex], cross through the origin, and continue to increase as it approaches the right asymptote at [latex]x=\\frac{\\pi}{2}[\/latex]. To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use<\/p>\n<div>\n<div style=\"text-align: center;\">[latex]f\\left(\\frac{P}{4}\\right)=A \\tan\\left(B\\frac{P}{4}\\right)=A\\tan\\left(B\\frac{\\pi}{4B}\\right)=A[\/latex]<\/div>\n<\/div>\n<p>because \u00a0[latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex].<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the function [latex]f(x)=A\\tan(Bx)[\/latex], graph one period.<\/strong><\/p>\n<ol>\n<li>Identify the stretching factor, |A|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Draw vertical asymptotes at \u00a0[latex]x=\u2212\\dfrac{P}{2}[\/latex] and [latex]x=\\frac{P}{2}[\/latex].<\/li>\n<li>For <em>A<\/em> &gt; 0 , the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for <em>A<\/em> &lt; 0 ).<\/li>\n<li>Plot reference points at [latex]\\left(\\frac{P}{4},A\\right)[\/latex]\u00a0(0, 0), and ([latex]\u2212\\dfrac{P}{4}[\/latex],\u2212 A), and draw the graph through these points.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Sketch a graph of one period of the function [latex]y=0.5\\tan\\left(\\frac{\\pi}{2}x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q302986\">Show Solution<\/button><\/p>\n<div id=\"q302986\" class=\"hidden-answer\" style=\"display: none\">First, we identify <em>A<\/em> and B.<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163807\/CNX_Precalc_Figure_06_02_002.jpg\" alt=\"An illustration of equations showing that A is the coefficient of tangent and B is the coefficient of x, which is within the tangent function.\" width=\"487\" height=\"113\" \/><\/p>\n<p>Because [latex]A=0.5[\/latex] and [latex]B=\\frac{\\pi}{2}[\/latex], we can find the <strong>stretching\/compressing factor<\/strong> and period. The period is [latex]\\frac{\\pi}{\\frac{\\pi}{2}}=2[\/latex], so the asymptotes are at [latex]x=\\pm 1[\/latex]. At a quarter period from the origin, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f(0.5)&=0.5\\tan\\left(\\frac{0.5\\pi}{2}\\right)\\\\ &=0.5\\tan(\\frac{\\pi}{4})\\\\ &=0.5 \\end{align}[\/latex]<\/p>\n<p>This means the curve must pass through the points(0.5,0.5),(0,0),and(\u22120.5,\u22120.5).The only inflection point is at the origin. Below is the graph of one period of the function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163810\/CNX_Precalc_Figure_06_02_003.jpg\" alt=\"A graph of one period of a modified tangent function, with asymptotes at x=-1 and x=1.\" width=\"487\" height=\"258\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>Sketch a graph of [latex]f(x)=3\\tan\\left(\\frac{\\pi}{6}x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q547078\">Show Solution<\/button><\/p>\n<div id=\"q547078\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163917\/CNX_Precalc_Figure_06_02_004.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm174880\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174880&theme=lumen&iframe_resize_id=ohm174880&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":6,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Animation: Graphing the Tangent Function Using the Unit Circle\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ssjG9kE25OY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":191,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"","project":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","license":"cc-by","license_terms":""},{"type":"copyrighted_video","description":"Animation: Graphing the Tangent Function Using the Unit Circle","author":"Mathispower4u","organization":"","url":"https:\/\/youtu.be\/ssjG9kE25OY","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/193"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/193\/revisions"}],"predecessor-version":[{"id":4636,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/193\/revisions\/4636"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/191"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/193\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=193"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=193"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=193"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}