{"id":167,"date":"2023-09-20T22:48:00","date_gmt":"2023-09-20T22:48:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/hyperbolic-functions\/"},"modified":"2024-08-05T12:30:24","modified_gmt":"2024-08-05T12:30:24","slug":"exponential-and-logarithmic-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/exponential-and-logarithmic-functions-learn-it-4\/","title":{"raw":"Exponential and Logarithmic Functions: Learn It 4","rendered":"Exponential and Logarithmic Functions: Learn It 4"},"content":{"raw":"

Hyperbolic Functions<\/h2>\r\n

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>\r\n

\r\n

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.\u00a0<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]\"A 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]\r\n\r\n

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>\r\n<\/section>\r\n

\r\n

hyperbolic functions<\/h3>\r\n

Hyperbolic cosine<\/strong><\/p>\r\n

[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\r\n

 <\/p>\r\n

Hyperbolic sine<\/strong><\/p>\r\n

[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\r\n

 <\/p>\r\n

Hyperbolic tangent<\/strong><\/p>\r\n

[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\r\n

 <\/p>\r\n

Hyperbolic cosecant<\/strong><\/p>\r\n

[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\r\n

 <\/p>\r\n

Hyperbolic secant<\/strong><\/p>\r\n

[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\r\n

 <\/p>\r\n

Hyperbolic cotangent<\/strong><\/p>\r\n

[latex]\\coth x=\\large \\frac{\\cosh x}{\\sinh x} \\normalsize = \\large \\frac{e^x+e^{\u2212x}}{e^x-e^{\u2212x}}[\/latex]<\/div>\r\n<\/section>\r\n
\r\n

The name cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name sinh<\/em> is pronounced \u201ccinch.\u201d Tanh<\/em>, sech<\/em>, csch<\/em>, and coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\r\n<\/section>\r\n

\r\n

But why are these functions called hyperbolic functions<\/em>?<\/p>\r\n

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>\r\n

[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>\r\n

 <\/p>\r\n

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>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]\"An Figure 7. The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].[\/caption]\r\n<\/section>\r\n

\r\n
\r\n

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>\r\n<\/section>\r\n

Graphs of Hyperbolic Functions<\/h3>\r\n

The graphs of [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex], can be derived by observing how they relate to exponential functions.<\/p>\r\n

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>\r\n

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>\r\n

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>\r\n

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\u00a0 [latex]-1[\/latex] as [latex]x[\/latex]\u00a0goes to positive or negative infinity, respectively.<\/p>\r\n

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>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"573\"]\"An Figure 8. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].[\/caption]\r\n<\/div>","rendered":"

Hyperbolic Functions<\/h2>\n

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

\n

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.\u00a0<\/p>\n

\"A
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

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

\n

hyperbolic functions<\/h3>\n

Hyperbolic cosine<\/strong><\/p>\n

[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\n

 <\/p>\n

Hyperbolic sine<\/strong><\/p>\n

[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\n

 <\/p>\n

Hyperbolic tangent<\/strong><\/p>\n

[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\n

 <\/p>\n

Hyperbolic cosecant<\/strong><\/p>\n

[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\n

 <\/p>\n

Hyperbolic secant<\/strong><\/p>\n

[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\n

 <\/p>\n

Hyperbolic cotangent<\/strong><\/p>\n

[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
\n

The name cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name sinh<\/em> is pronounced \u201ccinch.\u201d Tanh<\/em>, sech<\/em>, csch<\/em>, and coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\n<\/section>\n

\n

But why are these functions called hyperbolic functions<\/em>?<\/p>\n

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

[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>\n

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

\"An
Figure 7. The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].<\/figcaption><\/figure>\n<\/section>\n
\n
\n

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

Graphs of Hyperbolic Functions<\/h3>\n

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

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

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

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

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\u00a0 [latex]-1[\/latex] as [latex]x[\/latex]\u00a0goes to positive or negative infinity, respectively.<\/p>\n

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

\"An
Figure 8. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].<\/figcaption><\/figure>\n<\/div>\n","protected":false},"author":6,"menu_order":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.5 Exponential and Logarithmic Functions\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"1.5 Exponential and Logarithmic Functions","author":"","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/167"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":17,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/167\/revisions"}],"predecessor-version":[{"id":1282,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/167\/revisions\/1282"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/167\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=167"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=167"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=167"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}