Hyperbolic functions - Mathematics resources

1 Hyperbolic functionsThe Hyperbolic functions have similar names to the trigonmetric functions , but they are definedin terms of the exponential function. In this unit we define the three main Hyperbolic functions ,and sketch their graphs. We also discuss some identities relating these functions , and mentiontheir inverse functions and reciprocal order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second reading this text, and/or viewing the video tutorial on this topic, you should be able to: define the functionsf(x) = coshxandf(x) = sinhxin terms of the exponential function,and define the functionf(x) = tanhxin terms of coshxand sinhx, sketch the graphs of coshx, sinhxand tanhx, recognize the identities cosh2x sinh2x= 1 and sinh 2x= 2 sinhxcoshx, understand the meaning of the inverse functions sinh 1x, cosh 1xand tanh 1xand spec-ify their domains, define the reprocal functions sechx, cschxand (x) = (x) = (x) = for Hyperbolic related functions91c mathcentre January 9, 20061.

2 IntroductionIn this video we shall define the three Hyperbolic functionsf(x) = sinhx,f(x) = coshxandf(x) = tanhx. We shall look at the graphs of these functions , and investigate some of Definingf(x) = coshxThe Hyperbolic functions coshxand sinhxare defined using the exponential function ex. Weshall start with coshx. This is defined by the formulacoshx=ex+ e can use our knowledge of the graphs of exand e xto sketch the graph of coshx. First, letus calculate the value of cosh 0. Whenx= 0, ex= 1 and e x= 1. Socosh 0 =e0+ e 02=1 + 12= , let us see what happens asxgets large. We shall rewrite coshxascoshx=ex2+e see how this behaves asxgets large, recall the graphs of the two exponential xAsxgets larger, exincreases quickly, but e xdecreases quickly. So the second part of the sumex/2 + e x/2 gets very small asxgets large.

3 Therefore, asxgets larger, coshxgets closer andcloser to ex/2. We write this ascoshx ex2for the graph of coshxwill always stay above the graph of ex/2. This is because, even thoughe x/2 (the second part of the sum) gets very small, it is always greater than zero. Asxgetslarger and larger the difference between the two graphs gets smaller and mathcentre January 9, 20062 Now suppose thatx <0. Asxbecomes more negative, e xincreases quickly, but exdecreasesquickly, so the first part of the sum ex/2 + e x/2 gets very small. Asxgets more and morenegative, coshxgets closer and closer to e x/2. We write this ascoshx e x2for large , the graph of coshxwill always stay above the graph of e x/2 whenxis negative. This isbecause, even though ex/2 (the first part of the sum) gets very small, it is always greater thanzero.

4 But asxgets more and more negative the difference between the two graphs gets smallerand can now sketch the graph of coshx. Notice the graph is symmetric about they-axis, becausecoshx= cosh( x).yxcosh xKey PointThe Hyperbolic functionf(x) = coshxis defined by the formulacoshx=ex+ e function satisfies the conditions cosh 0 = 1 and coshx= cosh( x). The graph of coshxisalways above the graphs of ex/2 and e mathcentre January 9, 20063. Definingf(x) = sinhxWe shall now look at the Hyperbolic function sinhx. In speech, this function is pronounced as shine , or sometimes as sinch . The function is defined by the formulasinhx=ex e , we can use our knowledge of the graphs of exand e xto sketch the graph of sinhx. First,let us calculate the value of sinh 0. Whenx= 0, ex= 1 and e x= 1. Sosinh 0 =e0 e 02=1 12= , let us see what happens asxgets large.

5 We shall rewrite sinhxassinhx=ex2 e see how this behaves asxgets large, recall the graphs of the two exponential x Asxgets larger, exincreases quickly, but e xdecreases quickly. So the second part of thedifference ex/2 e x/2 gets very small asxgets large. Therefore, asxgets larger, sinhxgetscloser and closer to ex/2. We write this assinhx ex2for the graph of sinhxwill always stay below the graph ex/2. This is because, even though e x/2 (the second part of the difference) gets very small, it is always less than zero. Asxgetslarger and larger the difference between the two graphs gets smaller and mathcentre January 9, 20064 Next, suppose thatxis negative. As becomes more negative, e xbecomes large and negativevery quickly, butexdecreases very quickly. So asxbecomes more negative, the first part of thedifference ex/2 e x/2 gets very small.

6 So sinhxgets closer and closer to e x/2. We writethis assinhx e x2for large the graph of sinhxwill always stay above the graph of e x/2 whenxis negative. This isbecause, even though ex/2 (the first part of the difference) gets very small, it is always greaterthan zero. But asxgets more and more negative the difference between the two graphs getssmaller and can now sketch the graph of sinhx. Notice that sinh( x) = xKey PointThe Hyperbolic functionf(x) = sinhxis defined by the formulasinhx=ex e function satisfies the conditions sinh 0 = 0 and sinh( x) = sinhx. The graph of sinhxis always between the graphs of ex/2 and e mathcentre January 9, 2006We have seen that sinhxgets close to ex/2 asxgets large, and we have also seen that coshxgets close to ex/2 asxgets large. Therefore, sinhxand coshxmust get close together asxgetslarge.

7 Sosinhx coshxfor , we have seen that sinhxgets close to e x/2 asxgets large and negative, and wehave seen that coshxgets close to e x/2 asxgets large and negative. Therefore, sinhxand coshxmust get close together asxgets large and negative. Sosinhx coshxfor large can see this by sketching the graphs of sinhxand coshxon the same xcosh xKey PointFor large values ofxthe graphs of sinhxand coshxare close together. For large negativevalues ofxthe graphs of sinhxand coshxare close mathcentre January 9, 200664. Definingf(x) = tanhxWe shall now look at the Hyperbolic function tanhx. In speech, this function is pronounced as tansh , or sometimes as than . The function is defined by the formulatanhx= can work out tanhxout in terms of exponential functions . We know how sinhxand coshxare defined, so we can write tanhxastanhx=ex e x2 ex+ e x2=ex e xex+ e can use what we know about sinhxand coshxto sketch the graph of tanhx.

8 We first takex= 0. We know that sinh 0 = 0 and cosh 0 = 1, sotanh 0 =sinh 0cosh 0=01= large, sinhx coshx, so tanhxgets close to 1:tanhx 1 for sinhxis always less than coshx, so tanhxis always slightly less than 1. It gets close to 1asxgets very large, but never reaches large and negative, sinhx coshx, so tanhxgets close to 1:tanhx 1 for large sinhxis always greater than coshx, so tanhxis always slightly greater than 1. It getsclose to 1 asxgets very large and negative, but never reaches can now sketch the graph of tanhx. Notice that tanh( x) = x7c mathcentre January 9, 20065. Identities for Hyperbolic functionsHyperbolic functions have identities which are similar to,but not the same as, the identitiesfor trigonometric functions . In this section we shall provetwo of these identities, and list first identity iscosh2x sinh2x= prove this, we start by substituting the definitions for sinhxand coshx:cosh2x sinh2x=(ex+ e x2)2 (ex e x2) we expand the two squares in the numerators, we obtain(ex+ e x)2= e2x+ 2(ex)(e x) + e 2x= e2x+ 2 + e 2xand(ex e x)2= e2x 2(ex)(e x) + e 2x= e2x 2 + e 2x,where in each case we use the fact that (ex)(e x) = ex+( x)= e0= 1.

9 Using these expansions inour formula, we obtaincosh2x sinh2x=e2x+ 2 + e 2x4 e2x 2 + e we can move the factor of14out to the front, so thatcosh2x sinh2x=14((e2x+ 2 + e 2x) (e2x 2 + e 2x)).If, finally, we remove the inner brackets and simplify, we obtaincosh2x sinh2x=14(e2x+ 2 + e 2x e2x+ 2 e 2x)=14 4= 1,which is what we wanted to is another identity involving Hyperbolic functions :sinh 2x= 2 sinhxcoshx .On the left-hand side we have sinh 2xso, from the definition,sinh 2x=e2x e mathcentre January 9, 20068We want to manipulate the right-hand side to achieve this. Sowe shall start by substitutingthe definitions of sinhxand coshxinto the right-hand side:2 sinhxcoshx= 2(ex e x2) (ex+ e x2).We can cancel the 2 at the start with one of the 2 s in the denominator, and then we can takethe remaining factor of12out to the front.

10 We get2 sinhxcoshx=12(ex e x)(ex+ e x).Now we can multiply the two brackets together. This gives us2 sinhxcoshx=12(e2x+ 1 1 e 2x).Cancelling the ones finally gives us2 sinhxcoshx=12(e2x e 2x) = sinh 2x,which is what we wanted to are several more identities involving Hyperbolic functions :cosh 2x= (coshx)2+ (sinhx)2sinh(x+y) = sinhxcoshy+ sinhycoshxcosh(x+y) = coshxcoshy+ sinhxsinhycosh2x2=1 + coshx2sinh2x2=coshx 12If you know the trigonometric identities, you may notice that these Hyperbolic identities arevery similar, although sometimes plus signs have become minus signs and vice versa. In factthe Hyperbolic functions are very closely related to the trigonometric functions , and sinhxandcoshxare sometimes called the Hyperbolic sine and Hyperbolic cosine functions . If you go on tostudy complex numbers then you might learn more about how these functions are Other related functionsFinally, we shall look at some other functions that are related to the three Hyperbolic functionswe have just seen.