{"id":483,"date":"2025-02-13T19:45:22","date_gmt":"2025-02-13T19:45:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-background-youll-need-3\/"},"modified":"2025-02-13T19:45:22","modified_gmt":"2025-02-13T19:45:22","slug":"integration-of-exponential-logarithmic-and-hyperbolic-functions-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-background-youll-need-3\/","title":{"raw":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Background You'll Need 3","rendered":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Background You&#8217;ll Need 3"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Identify hyperbolic functions their graphs, and understand their fundamental identities<\/li>\n<\/ul>\n<\/section>\n<h2>Hyperbolic Functions<\/h2>\n<p><strong>Hyperbolic functions<\/strong> are defined in terms of certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes.<\/p>\n<section class=\"textbox example\">\n<p>Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary. If we introduce a coordinate system so that the low point of the chain lies along the [latex]y[\/latex]-axis, we can describe the height of the chain in terms of a hyperbolic function.&nbsp;<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202700\/CNX_Calc_Figure_01_05_009.jpg\" alt=\"A photograph of a spider web collecting dew drops.\" width=\"488\" height=\"403\"> Figure 6. The shape of a strand of silk in a spider\u2019s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: \u201cMtpaley\u201d, Wikimedia Commons)[\/caption]\n\n<p>Using the definition of [latex]\\cosh(x)[\/latex] and principles of physics, it can be shown that the height of a hanging chain can be described by the function [latex]h(x)=a \\cosh(x\/a)+c[\/latex] for certain constants [latex]a[\/latex] and [latex]c[\/latex].<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>hyperbolic functions<\/h3>\n<p id=\"fs-id1170572467989\"><strong>Hyperbolic cosine<\/strong><\/p>\n<div id=\"fs-id1170572467996\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572468030\"><strong>Hyperbolic sine<\/strong><\/p>\n<div id=\"fs-id1170572468036\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572468070\"><strong>Hyperbolic tangent<\/strong><\/p>\n<div id=\"fs-id1170572468077\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431436\"><strong>Hyperbolic cosecant<\/strong><\/p>\n<div id=\"fs-id1170572431443\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431488\"><strong>Hyperbolic secant<\/strong><\/p>\n<div id=\"fs-id1170572431494\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431539\"><strong>Hyperbolic cotangent<\/strong><\/p>\n<div id=\"fs-id1170572431545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\coth x=\\large \\frac{\\cosh x}{\\sinh x} \\normalsize = \\large \\frac{e^x+e^{\u2212x}}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>The name <em>cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name <em>sinh<\/em> is pronounced \u201ccinch.\u201d <em>Tanh<\/em>, <em>sech<\/em>, <em>csch<\/em>, and <em>coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p id=\"fs-id1170572234364\">But why are these functions called <em>hyperbolic functions<\/em>?<\/p>\n<p>To answer this question, consider the quantity [latex]\\cosh^2 t-\\sinh^2 t[\/latex]. Using the definition of [latex]\\cosh[\/latex] and [latex]\\sinh[\/latex], we see that<\/p>\n<div id=\"fs-id1170572234411\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 t-\\sinh^2 t=\\large \\frac{e^{2t}+2+e^{-2t}}{4}-\\frac{e^{2t}-2+e^{-2t}}{4} \\normalsize =1[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572171610\">This identity is the analog of the trigonometric identity [latex]\\cos^2 t+\\sin^2 t=1[\/latex]. Here, given a value [latex]t[\/latex], the point [latex](x,y)=(\\cosh t,\\sinh t)[\/latex] lies on the unit hyperbola [latex]x^2-y^2=1[\/latex] (Figure 7).<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202702\/CNX_Calc_Figure_01_05_007.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation \u201c(x squared) - (y squared) -1\u201d. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways \u201cU\u201d shape. There is a point plotted on the graph of the relation labeled \u201c(cosh(1), sinh(1))\u201d, which is at the approximate point (1.5, 1.2).\" width=\"325\" height=\"275\"> Figure 7. The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].[\/caption]\n<\/section>\n<div id=\"fs-id1170572171758\" class=\"bc-section section\">\n<section class=\"textbox proTip\">\n<p>If you think hyperbolic functions look a lot like trigonometric ones, you're not wrong! They share similar properties because they're both connected to the concept of the exponential function [latex]e^x[\/latex]. Remember, while trigonometric functions relate to the unit circle, hyperbolic functions are associated with the unit hyperbola.<\/p>\n<\/section>\n<h3>Graphs of Hyperbolic Functions<\/h3>\n<p id=\"fs-id1170572171763\">The graphs of [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex], can be derived by observing how they relate to exponential functions.<\/p>\n<p>As [latex]x[\/latex] approaches towards infinity, both functions approach [latex]\\frac{1}{2}e^x[\/latex] because the term [latex]e^{\u2212x}[\/latex] becomes negligible.<\/p>\n<p>In contrast, as [latex]x[\/latex] moves towards negative infinity, [latex]\\cosh x[\/latex] mirrors [latex]\\frac{1}{2}e^{\u2212x}[\/latex], while [latex]\\sinh x[\/latex] mirrors [latex]-\\frac{1}{2}e^{\u2212x}[\/latex].<\/p>\n<p>Therefore, the graphs [latex]\\frac{1}{2}e^x, \\, \\frac{1}{2}e^{\u2212x}[\/latex], and [latex]\u2212\\frac{1}{2}e^{\u2212x}[\/latex] provide a roadmap for sketching the graphs.<\/p>\n<p>When graphing [latex]\\tanh x[\/latex], we note that its value starts at [latex]0[\/latex] when [latex]x[\/latex] is [latex]0[\/latex] and then ascends towards [latex]1[\/latex] or descends towards&nbsp; [latex]-1[\/latex] as [latex]x[\/latex]&nbsp;goes to positive or negative infinity, respectively.<\/p>\n<p>The graphs of the other three hyperbolic functions can be sketched using the graphs of [latex]\\cosh x, \\, \\sinh x[\/latex], and [latex]\\tanh x[\/latex] (Figure 8).<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"573\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202706\/CNX_Calc_Figure_01_05_011.jpg\" alt=\"An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function \u201cy = cosh(x)\u201d, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of -x)\u201d, a decreasing curved function and the second of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function. The function \u201cy = cosh(x)\u201d is always above these two functions without ever touching them. The second graph is of the function \u201cy = sinh(x)\u201d, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function and the second of these functions is \u201cy = -(1\/2)(e to power of -x)\u201d, an increasing curved function that approaches the x axis without touching it. The function \u201cy = sinh(x)\u201d is always between these two functions without ever touching them. The third graph is of the function \u201cy = sech(x)\u201d, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function \u201cy = csch(x)\u201d. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function \u201cy = tanh(x)\u201d, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = 1\u201d, a horizontal line function and the second of these functions is \u201cy = -1\u201d, another horizontal line function. The function \u201cy = tanh(x)\u201d is always between these two functions without ever touching them. The sixth graph is of the function \u201cy = coth(x)\u201d. On the left side of the y axis, the function starts slightly below the boundary line \u201cy = 1\u201d and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line \u201cy = -1\u201d, which it never touches.\" width=\"573\" height=\"929\"> Figure 8. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].[\/caption]\n<\/div>\n<h3>Identities Involving Hyperbolic Functions<\/h3>\n<p>Just as trigonometric functions have identities that allow for the simplification and transformation of expressions, hyperbolic functions also possess their own set of identities.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>hyperbolic function identities<\/h3>\n<div><strong>Hyperbolic Reflection Identities:<\/strong>\n<ul>\n\t<li>[latex]\\cosh(\u2212x)=\\cosh x[\/latex]<\/li>\n\t<li>[latex]\\sinh(\u2212x)=\u2212\\sinh x[\/latex]<\/li>\n<\/ul>\n<strong>Hyperbolic Pythagorean Identities:<\/strong>\n<ul>\n\t<li>[latex]\\cosh^2 x-\\sinh^2 x=1[\/latex]<\/li>\n<\/ul>\n<p><strong>Hyperbolic Squared Identities:<\/strong><\/p>\n<ul>\n\t<li>[latex]1-\\tanh^2 x=\\text{sech}^2 x[\/latex]<\/li>\n\t<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\n<\/ul>\n<strong>Hyperbolic Addition Formulas:<\/strong>\n<ul>\n\t<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\n\t<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\n<\/ul>\n<strong>Exponential Definitions<\/strong> <strong>of Hyperbolic Functions<\/strong><br>\n<ul>\n\t<li>[latex]\\cosh x+\\sinh x=e^x[\/latex]<\/li>\n\t<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<ol id=\"fs-id1170572443403\" style=\"list-style-type: lower-alpha;\">\n\t<li>Simplify [latex]\\sinh(5 \\ln x)[\/latex].<\/li>\n\t<li>If [latex]\\sinh x=\\frac{3}{4}[\/latex], find the values of the remaining five hyperbolic functions.<\/li>\n<\/ol>\n<p>[reveal-answer q=\"fs-id1170572443462\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1170572443462\"]<\/p>\n<ol id=\"fs-id1170572443462\" style=\"list-style-type: lower-alpha;\">\n\t<li>Using the definition of the [latex]\\sinh[\/latex] function, we write<br>\n<div id=\"fs-id1170570995857\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh(5 \\ln x)=\\large \\frac{e^{5 \\ln x}-e^{-5 \\ln x}}{2} \\normalsize = \\large \\frac{e^{\\ln(x^5)}-e^{\\ln(x^{-5})}}{2} \\normalsize =\\large \\frac{x^5-x^{-5}}{2}[\/latex].<\/div>\n<\/li>\n\t<li>Using the identity [latex]\\cosh^2 x-\\sinh^2 x=1[\/latex], we see that<br>\n<div id=\"fs-id1170573388429\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 x=1+\\big(\\frac{3}{4}\\big)^2=\\frac{25}{16}[\/latex].<\/div>\n<p>Since [latex]\\cosh x \\ge 1[\/latex] for all [latex]x[\/latex], we must have [latex]\\cosh x=5\/4[\/latex]. Then, using the definitions for the other hyperbolic functions, we conclude that [latex]\\tanh x=3\/5, \\, \\text{csch} \\, x=4\/3, \\, \\text{sech} \\, x=4\/5[\/latex], and [latex]\\coth x=5\/3[\/latex].<\/p>\n<\/li>\n<\/ol>\n\nWatch the following video to see the worked solution to this example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=1498&amp;end=1738&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\n\n<p>You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.5ExponentialAndLogarithmicFunctions1498to1738_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]287174[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify hyperbolic functions their graphs, and understand their fundamental identities<\/li>\n<\/ul>\n<\/section>\n<h2>Hyperbolic Functions<\/h2>\n<p><strong>Hyperbolic functions<\/strong> are defined in terms of certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes.<\/p>\n<section class=\"textbox example\">\n<p>Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary. If we introduce a coordinate system so that the low point of the chain lies along the [latex]y[\/latex]-axis, we can describe the height of the chain in terms of a hyperbolic function.&nbsp;<\/p>\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202700\/CNX_Calc_Figure_01_05_009.jpg\" alt=\"A photograph of a spider web collecting dew drops.\" width=\"488\" height=\"403\" \/><figcaption class=\"wp-caption-text\">Figure 6. The shape of a strand of silk in a spider\u2019s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: \u201cMtpaley\u201d, Wikimedia Commons)<\/figcaption><\/figure>\n<p>Using the definition of [latex]\\cosh(x)[\/latex] and principles of physics, it can be shown that the height of a hanging chain can be described by the function [latex]h(x)=a \\cosh(x\/a)+c[\/latex] for certain constants [latex]a[\/latex] and [latex]c[\/latex].<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>hyperbolic functions<\/h3>\n<p id=\"fs-id1170572467989\"><strong>Hyperbolic cosine<\/strong><\/p>\n<div id=\"fs-id1170572467996\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572468030\"><strong>Hyperbolic sine<\/strong><\/p>\n<div id=\"fs-id1170572468036\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572468070\"><strong>Hyperbolic tangent<\/strong><\/p>\n<div id=\"fs-id1170572468077\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431436\"><strong>Hyperbolic cosecant<\/strong><\/p>\n<div id=\"fs-id1170572431443\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431488\"><strong>Hyperbolic secant<\/strong><\/p>\n<div id=\"fs-id1170572431494\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431539\"><strong>Hyperbolic cotangent<\/strong><\/p>\n<div id=\"fs-id1170572431545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\coth x=\\large \\frac{\\cosh x}{\\sinh x} \\normalsize = \\large \\frac{e^x+e^{\u2212x}}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>The name <em>cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name <em>sinh<\/em> is pronounced \u201ccinch.\u201d <em>Tanh<\/em>, <em>sech<\/em>, <em>csch<\/em>, and <em>coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p id=\"fs-id1170572234364\">But why are these functions called <em>hyperbolic functions<\/em>?<\/p>\n<p>To answer this question, consider the quantity [latex]\\cosh^2 t-\\sinh^2 t[\/latex]. Using the definition of [latex]\\cosh[\/latex] and [latex]\\sinh[\/latex], we see that<\/p>\n<div id=\"fs-id1170572234411\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 t-\\sinh^2 t=\\large \\frac{e^{2t}+2+e^{-2t}}{4}-\\frac{e^{2t}-2+e^{-2t}}{4} \\normalsize =1[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572171610\">This identity is the analog of the trigonometric identity [latex]\\cos^2 t+\\sin^2 t=1[\/latex]. Here, given a value [latex]t[\/latex], the point [latex](x,y)=(\\cosh t,\\sinh t)[\/latex] lies on the unit hyperbola [latex]x^2-y^2=1[\/latex] (Figure 7).<\/p>\n<figure style=\"width: 325px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202702\/CNX_Calc_Figure_01_05_007.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation \u201c(x squared) - (y squared) -1\u201d. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways \u201cU\u201d shape. There is a point plotted on the graph of the relation labeled \u201c(cosh(1), sinh(1))\u201d, which is at the approximate point (1.5, 1.2).\" width=\"325\" height=\"275\" \/><figcaption class=\"wp-caption-text\">Figure 7. The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].<\/figcaption><\/figure>\n<\/section>\n<div id=\"fs-id1170572171758\" class=\"bc-section section\">\n<section class=\"textbox proTip\">\n<p>If you think hyperbolic functions look a lot like trigonometric ones, you&#8217;re not wrong! They share similar properties because they&#8217;re both connected to the concept of the exponential function [latex]e^x[\/latex]. Remember, while trigonometric functions relate to the unit circle, hyperbolic functions are associated with the unit hyperbola.<\/p>\n<\/section>\n<h3>Graphs of Hyperbolic Functions<\/h3>\n<p id=\"fs-id1170572171763\">The graphs of [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex], can be derived by observing how they relate to exponential functions.<\/p>\n<p>As [latex]x[\/latex] approaches towards infinity, both functions approach [latex]\\frac{1}{2}e^x[\/latex] because the term [latex]e^{\u2212x}[\/latex] becomes negligible.<\/p>\n<p>In contrast, as [latex]x[\/latex] moves towards negative infinity, [latex]\\cosh x[\/latex] mirrors [latex]\\frac{1}{2}e^{\u2212x}[\/latex], while [latex]\\sinh x[\/latex] mirrors [latex]-\\frac{1}{2}e^{\u2212x}[\/latex].<\/p>\n<p>Therefore, the graphs [latex]\\frac{1}{2}e^x, \\, \\frac{1}{2}e^{\u2212x}[\/latex], and [latex]\u2212\\frac{1}{2}e^{\u2212x}[\/latex] provide a roadmap for sketching the graphs.<\/p>\n<p>When graphing [latex]\\tanh x[\/latex], we note that its value starts at [latex]0[\/latex] when [latex]x[\/latex] is [latex]0[\/latex] and then ascends towards [latex]1[\/latex] or descends towards&nbsp; [latex]-1[\/latex] as [latex]x[\/latex]&nbsp;goes to positive or negative infinity, respectively.<\/p>\n<p>The graphs of the other three hyperbolic functions can be sketched using the graphs of [latex]\\cosh x, \\, \\sinh x[\/latex], and [latex]\\tanh x[\/latex] (Figure 8).<\/p>\n<figure style=\"width: 573px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202706\/CNX_Calc_Figure_01_05_011.jpg\" alt=\"An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function \u201cy = cosh(x)\u201d, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of -x)\u201d, a decreasing curved function and the second of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function. The function \u201cy = cosh(x)\u201d is always above these two functions without ever touching them. The second graph is of the function \u201cy = sinh(x)\u201d, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function and the second of these functions is \u201cy = -(1\/2)(e to power of -x)\u201d, an increasing curved function that approaches the x axis without touching it. The function \u201cy = sinh(x)\u201d is always between these two functions without ever touching them. The third graph is of the function \u201cy = sech(x)\u201d, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function \u201cy = csch(x)\u201d. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function \u201cy = tanh(x)\u201d, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = 1\u201d, a horizontal line function and the second of these functions is \u201cy = -1\u201d, another horizontal line function. The function \u201cy = tanh(x)\u201d is always between these two functions without ever touching them. The sixth graph is of the function \u201cy = coth(x)\u201d. On the left side of the y axis, the function starts slightly below the boundary line \u201cy = 1\u201d and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line \u201cy = -1\u201d, which it never touches.\" width=\"573\" height=\"929\" \/><figcaption class=\"wp-caption-text\">Figure 8. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].<\/figcaption><\/figure>\n<\/div>\n<h3>Identities Involving Hyperbolic Functions<\/h3>\n<p>Just as trigonometric functions have identities that allow for the simplification and transformation of expressions, hyperbolic functions also possess their own set of identities.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>hyperbolic function identities<\/h3>\n<div><strong>Hyperbolic Reflection Identities:<\/strong><\/p>\n<ul>\n<li>[latex]\\cosh(\u2212x)=\\cosh x[\/latex]<\/li>\n<li>[latex]\\sinh(\u2212x)=\u2212\\sinh x[\/latex]<\/li>\n<\/ul>\n<p><strong>Hyperbolic Pythagorean Identities:<\/strong><\/p>\n<ul>\n<li>[latex]\\cosh^2 x-\\sinh^2 x=1[\/latex]<\/li>\n<\/ul>\n<p><strong>Hyperbolic Squared Identities:<\/strong><\/p>\n<ul>\n<li>[latex]1-\\tanh^2 x=\\text{sech}^2 x[\/latex]<\/li>\n<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\n<\/ul>\n<p><strong>Hyperbolic Addition Formulas:<\/strong><\/p>\n<ul>\n<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\n<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\n<\/ul>\n<p><strong>Exponential Definitions<\/strong> <strong>of Hyperbolic Functions<\/strong><\/p>\n<ul>\n<li>[latex]\\cosh x+\\sinh x=e^x[\/latex]<\/li>\n<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<ol id=\"fs-id1170572443403\" style=\"list-style-type: lower-alpha;\">\n<li>Simplify [latex]\\sinh(5 \\ln x)[\/latex].<\/li>\n<li>If [latex]\\sinh x=\\frac{3}{4}[\/latex], find the values of the remaining five hyperbolic functions.<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572443462\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572443462\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572443462\" style=\"list-style-type: lower-alpha;\">\n<li>Using the definition of the [latex]\\sinh[\/latex] function, we write\n<div id=\"fs-id1170570995857\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh(5 \\ln x)=\\large \\frac{e^{5 \\ln x}-e^{-5 \\ln x}}{2} \\normalsize = \\large \\frac{e^{\\ln(x^5)}-e^{\\ln(x^{-5})}}{2} \\normalsize =\\large \\frac{x^5-x^{-5}}{2}[\/latex].<\/div>\n<\/li>\n<li>Using the identity [latex]\\cosh^2 x-\\sinh^2 x=1[\/latex], we see that\n<div id=\"fs-id1170573388429\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 x=1+\\big(\\frac{3}{4}\\big)^2=\\frac{25}{16}[\/latex].<\/div>\n<p>Since [latex]\\cosh x \\ge 1[\/latex] for all [latex]x[\/latex], we must have [latex]\\cosh x=5\/4[\/latex]. Then, using the definitions for the other hyperbolic functions, we conclude that [latex]\\tanh x=3\/5, \\, \\text{csch} \\, x=4\/3, \\, \\text{sech} \\, x=4\/5[\/latex], and [latex]\\coth x=5\/3[\/latex].<\/p>\n<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=1498&amp;end=1738&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.5ExponentialAndLogarithmicFunctions1498to1738_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm287174\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287174&theme=lumen&iframe_resize_id=ohm287174&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":479,"module-header":"","content_attributions":[],"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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/483"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/483\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/479"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/483\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=483"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=483"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=483"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=483"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}