Inverse Hyperbolic Functions

Total Page：16

File Type：pdf, Size：1020Kb

INVERSE HYPERBOLIC FUNCTIONS We will recall that the function eex − − x sinh x = 2 is a 1 to 1 mapping.ie one value of x maps onto one value of sinhx. This implies that the inverse function exists. y =⇔=sinhxx sinh−1 y The function f(x)=sinhx clearly involves the exponential function. We will now find the inverse function in its logarithmic form. LET y = sinh−1 x x = sinh y THEN eey − − y x = 2 (BY DEFINITION) SOME ALGEBRAIC MANIPULATION GIVES yy− 2xe=− e MULTIPLYING BOTH SIDES BY e y yy2 21xe= e − RE-ARRANGING TO CREATE A QUADRATIC IN e y 2 yy 02=−exe −1 APPLYING THE QUADRATIC FORMULA 24xx± 2 + 4 e y = 2 22xx± 2 + 1 e y = 2 y 2 exx= ±+1 SINCE y e >0 WE MUST ONLY CONSIDER THE ADDITION. y 2 exx= ++1 TAKING LOGARITHMS OF BOTH SIDES yxx=++ln2 1 { } IN CONCLUSION −12 sinhxxx=++ ln 1 { } THIS IS CALLED THE LOGARITHMIC FORM OF THE INVERSE FUNCTION. The graph of the inverse function is of course a reflection in the line y=x. THE INVERSE OF COSHX The function eex + − x fx()== cosh x 2 Is of course not a one to one function. (observe the graph of the function earlier). In order to have an inverse the DOMAIN of the function is restricted. eexx+ − fx()== cosh x , x≥ 0 2 This is a one to one mapping with RANGE [1, ∞] . We find the inverse function in logarithmic form in a similar way. y = cosh −1 x x = cosh y provided y ≥ 0 eeyy+ − xy==cosh 2 2xe=+yy e− 02=−+exeyy− 2 yy 02=−exe +1 USING QUADRATIC FORMULA −24xx±−2 4 e y = 2 y 2 exx=± −1 TAKING LOGS yxx=±−ln2 1 { } yxx=+−ln2 1 is one possible expression { } APPROPRIATE IF THE DOMAIN OF THE ORIGINAL FUNCTION HAD BEEN RESTRICTED AS EXPLAINED ABOVE. HOWEVER CONSIDERING THE WHOLE FUNCTION (UNRESTRICTED) FOR EACH VALUE OF X THERE ARE TWO EQUAL AND OPPOSITE VALUES OF Y. This implies cosh−12xxx= ±+− ln 1 { } To add a more sound mathematical argument to this we can use some algebraic manipulation and laws of logs to the alternative solution. Consider (1)(xx− 22−+− xx1) xx−−≡2 1 (1)xx+−2 THIS IS NOTHING BUT A USEFUL TRICK! The numerator becomes 1 (check it and see!) So 1 xx−−≡221( ≡+−xx1)−1 (1)xx+−2 Now returning to out alternative solution −12 yxxx==±+cosh ln− 1 { } Covers all possible solutions depending on the restriction of the domain TASK: Show that the logarithmic form of the hyperbolic tan is −1 1 1+ x y ==tanhxx ln for ∠1 HINT 2 1− x Start as before by rearranging the inverse statement and then use eex − −x tanh x = eex + −x THE GOOD NEWS! THE FORMULAE FOR THE LOGARITHMIC FORMS OF INVERSE HYPERBOLIC FUNCTIONS ARE IN THE WJEC FORMULA BOOK! .

Copy Link

Recommended publications

Section 6.7 Hyperbolic Functions 3
Section 6.7 Difference Equations to Hyperbolic Functions Differential Equations The final class of functions we will consider are the hyperbolic functions. In a sense these functions are not new to us since they may all be expressed in terms of the exponential function and its inverse, the natural logarithm function. However, we will see that they have many interesting and useful properties. Definition For any real number x, the hyperbolic sine of x, denoted sinh(x), is defined by 1 sinh(x) = (ex − e−x) (6.7.1) 2 and the hyperbolic cosine of x, denoted cosh(x), is defined by 1 cosh(x) = (ex + e−x). (6.7.2) 2 Note that, for any real number t, 1 1 cosh2(t) − sinh2(t) = (et + e−t)2 − (et − e−t)2 4 4 1 1 = (e2t + 2ete−t + e−2t) − (e2t − 2ete−t + e−2t) 4 4 1 = (2 + 2) 4 = 1. Thus we have the useful identity cosh2(t) − sinh2(t) = 1 (6.7.3) for any real number t. Put another way, (cosh(t), sinh(t)) is a point on the hyperbola x2 −y2 = 1. Hence we see an analogy between the hyperbolic cosine and sine functions and the cosine and sine functions: Whereas (cos(t), sin(t)) is a point on the circle x2 + y2 = 1, (cosh(t), sinh(t)) is a point on the hyperbola x2 − y2 = 1. In fact, the cosine and sine functions are sometimes referred to as the circular cosine and sine functions. We shall see many more similarities between the hyperbolic trigonometric functions and their circular counterparts as we proceed with our discussion.
[Show full text]
“An Introduction to Hyperbolic Functions in Elementary Calculus” Jerome Rosenthal, Broward Community College, Pompano Beach, FL 33063
“An Introduction to Hyperbolic Functions in Elementary Calculus” Jerome Rosenthal, Broward Community College, Pompano Beach, FL 33063 Mathematics Teacher, April 1986, Volume 79, Number 4, pp. 298–300. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). The primary purpose of the National Council of Teachers of Mathematics is to provide vision and leadership in the improvement of the teaching and learning of mathematics. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprint with permission from Mathematics Teacher, copyright April 1986 by the National Council of Teachers of Mathematics. All rights reserved. he customary introduction to hyperbolic functions mentions that the combinations T s1y2dseu 1 e2ud and s1y2dseu 2 e2ud occur with sufficient frequency to warrant special names. These functions are analogous, respectively, to cos u and sin u. This article attempts to give a geometric justification for cosh and sinh, comparable to the functions of sin and cos as applied to the unit circle. This article describes a means of identifying cosh u 5 s1y2dseu 1 e2ud and sinh u 5 s1y2dseu 2 e2ud with the coordinates of a point sx, yd on a comparable “unit hyperbola,” x2 2 y2 5 1. The functions cosh u and sinh u are the basic hyperbolic functions, and their relationship to the so-called unit hyperbola is our present concern. The basic hyperbolic functions should be presented to the student with some rationale.
[Show full text]
2 COMPLEX NUMBERS 61 2 Complex Numbers
2 COMPLEX NUMBERS 61 2 Complex numbers Though they don’t exist, per se, complex numbers pop up all over the place when we do science. They are an especially useful mathematical tool in problems dealing with oscillations and waves in light, fluids, magnetic fields, electrical circuits, etc., • stability problems in fluid flow and structural engineering, • signal processing (Fourier transform, etc.), • quantum physics, e.g. Schroedinger’s equation, • ∂ψ i + 2ψ + V (x)ψ =0 , ∂t ∇ which describes the wavefunctions of atomic and molecular systems,. difficult differential equations. • For your revision: recall that the general quadratic equation, ax2 + bx + c =0, has two solutions given by the quadratic formula b √b2 4ac x = − ± − . (85) 2a But what happens when the discriminant is negative? For instance, consider x2 2x +2=0 . (86) − Its solutions are 2 ( 2)2 4 1 2 2 √ 4 x = ± − − × × = ± − 2 1 2 p × = 1 √ 1 . ± − 2 COMPLEX NUMBERS 62 These two solutions are evidently not real numbers! However, if we put that aside and accept the existence of the square root of minus one, written as i = √ 1 , (87) − then the two solutions, 1+ i and 1 i, live in the set of complex numbers. The − quadratic equation now can be factorised as z2 2z +2=(z 1 i)(z 1+ i)=0 , − − − − and to be able to factorise every polynomial is a very useful thing, as we shall see. 2.1 Complex algebra 2.1.1 Definitions A complex number z takes the form z = x + iy , (88) where x and y are real numbers and i is the imaginary unit satisfying i2 = 1 .
[Show full text]
Hyperbolic Functions Certain Combinations of the Exponential
Hyperbolic Functions Certain combinations of the exponential function occur so often in physical applications that they are given special names. Specifically, half the difference of ex and e−x is defined as the hyperbolic sine function and half their sum is the hyperbolic cosine function. These functions are denoted as follows: ex − e−x ex + e−x sinh x = and cosh x = . (1.1) 2 2 The names of the hyperbolic functions and their notations bear a striking re- semblance to those for the trigonometric functions, and there are reasons for this. First, the hyperbolic functions sinh x and cosh x are related to the curve x2 −y2 = 1, called the unit hyperbola, in much the same way as the trigonometric functions sin x and cos x are related to the unit circle x2 + y2 = 1. Second, for each identity satis- fied by the trigonometric functions, there is a corresponding identity satisfied by the hyperbolic functions — not the same identity, but one very similar. For example, using equations 1.1, we have ex + e−x 2 ex − e−x 2 (cosh x)2 − (sinh x)2 = − 2 2 1 x − x x − x = e2 +2+ e 2 − e2 − 2+ e 2 4 =1. Thus the hyperbolic sine and cosine functions satisfy the identity cosh2 x − sinh2 x =1, (1.2) which is reminiscent of the identity cos2 x + sin2 x = 1 for the trigonometric func- tions. Just as four other trigonometric functions are defined in terms of sin x and cos x, four corresponding hyperbolic functions are defined as follows: sinh x ex − e−x cosh x ex + e−x tanh x = = , cothx = = , cosh x ex + e−x sinh x ex − e−x (1.3) 1 2 1 2 sechx = = , cschx = = .
[Show full text]
Accurate Hyperbolic Tangent Computation
Accurate Hyperbolic Tangent Computation Nelson H. F. Beebe Center for Scientiﬁc Computing Department of Mathematics University of Utah Salt Lake City, UT 84112 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 Internet: [email protected] 20 April 1993 Version 1.07 Abstract These notes for Mathematics 119 describe the Cody-Waite algorithm for ac- curate computation of the hyperbolic tangent, one of the simpler elementary functions. Contents 1 Introduction 1 2 The plan of attack 1 3 Properties of the hyperbolic tangent 1 4 Identifying computational regions 3 4.1 Finding xsmall ::::::::::::::::::::::::::::: 3 4.2 Finding xmedium ::::::::::::::::::::::::::: 4 4.3 Finding xlarge ::::::::::::::::::::::::::::: 5 4.4 The rational polynomial ::::::::::::::::::::::: 6 5 Putting it all together 7 6 Accuracy of the tanh() computation 8 7 Testing the implementation of tanh() 9 List of Figures 1 Hyperbolic tangent, tanh(x). :::::::::::::::::::: 2 2 Computational regions for evaluating tanh(x). ::::::::::: 4 3 Hyperbolic cosecant, csch(x). :::::::::::::::::::: 10 List of Tables 1 Expected errors in polynomial evaluation of tanh(x). ::::::: 8 2 Relative errors in tanh(x) on Convex C220 (ConvexOS 9.0). ::: 15 3 Relative errors in tanh(x) on DEC MicroVAX 3100 (VMS 5.4). : 15 4 Relative errors in tanh(x) on Hewlett-Packard 9000/720 (HP-UX A.B8.05). ::::::::::::::::::::::::::::::: 15 5 Relative errors in tanh(x) on IBM 3090 (AIX 370). :::::::: 15 6 Relative errors in tanh(x) on IBM RS/6000 (AIX 3.1). :::::: 16 7 Relative errors in tanh(x) on MIPS RC3230 (RISC/os 4.52). :: 16 8 Relative errors in tanh(x) on NeXT (Motorola 68040, Mach 2.1).
[Show full text]
Mathematics Tables
MATHEMATICS TABLES Department of Applied Mathematics Naval Postgraduate School TABLE OF CONTENTS Derivatives and Diﬀerentials.............................................. 1 Integrals of Elementary Forms............................................ 2 Integrals Involving au + b ................................................ 3 Integrals Involving u2 a2 ............................................... 4 Integrals Involving √u±2 a2, a> 0 ....................................... 4 Integrals Involving √a2 ± u2, a> 0 ....................................... 5 Integrals Involving Trigonometric− Functions .............................. 6 Integrals Involving Exponential Functions ................................ 7 Miscellaneous Integrals ................................................... 7 Wallis’ Formulas ................................................... ...... 8 Gamma Function ................................................... ..... 8 Laplace Transforms ................................................... 9 Probability and Statistics ............................................... 11 Discrete Probability Functions .......................................... 12 Standard Normal CDF and Table ....................................... 12 Continuous Probability Functions ....................................... 13 Fourier Series ................................................... ........ 13 Separation of Variables .................................................. 15 Bessel Functions ................................................... ..... 16 Legendre
[Show full text]
5.8 Hyperbolic Functions 383
5.8 Hyperbolic Functions 383 5.8 Hyperbolic Functions Develop properties of hyperbolic functions. Differentiate and integrate hyperbolic functions. Develop properties of inverse hyperbolic functions. Differentiate and integrate functions involving inverse hyperbolic functions. Hyperbolic Functions In this section, you will look briefly at a special class of exponential functions called hyperbolic functions. The name hyperbolic function arose from comparison of the area of a semicircular region, as shown in Figure 5.29, with the area of a region under a hyperbola, as shown in Figure 5.30. y y y = 1 + x2 2 2 y = 1 − x2 x x JOHANN HEINRICH LAMBERT −11−11 (1728–1777) 2 ϩ 2 ϭ Ϫ 2 ϩ 2 ϭ The first person to publish a Circle:x y 1 Hyperbola: x y 1 comprehensive study on hyperbolic Figure 5.29 Figure 5.30 functions was Johann Heinrich Lambert, a Swiss-German mathe- The integral for the semicircular region involves an inverse trigonometric (circular) matician and colleague of Euler. See LarsonCalculus.com to read function: more of this biography. 1 1 1 ␲ ͵ Ί1 Ϫ x2 dx ϭ ΄xΊ1 Ϫ x2 ϩ arcsin x ΅ ϭ Ϸ 1.571. Ϫ1 2 Ϫ1 2 The integral for the hyperbolic region involves an inverse hyperbolic function: 1 1 1 ͵ Ί ϩ 2 ϭ Ί ϩ 2 ϩ Ϫ1 Ϸ 1 x dx ΄x 1 x sinh x ΅ 2.296. Ϫ1 2 Ϫ1 This is only one of many ways in which the hyperbolic functions are similar to the trigonometric functions. Definitions of the Hyperbolic Functions REMARK The notation e x Ϫ eϪx 1 sinh x ϭ csch x ϭ , x 0 sinh x is read as “the hyperbolic 2 sinh x sine of x,” cosh x as “the e x ϩ eϪx 1 cosh x ϭ sech x ϭ hyperbolic cosine of x,” and 2 cosh x so on.
[Show full text]
Hyperbolic Functions
Hyperbolic Functions The functions hyperbolic sine and hyperbolic cosine, written, respectively as sinh and cosh, are well known functions defined by the formulae ex − e−x ex + e−x sinh (x) := ; and cosh (x) := ; 2 2 were first studied by Riccati in the mid-18th century. He applied them to the solution of general quadratic equations with real coefficients and he found a number of the standard identities that are similar to those familiar from the study of sine and consine functions. The functions have a number of applica- tions. For one, they were crucial in the development of navigational charts using Mercator Projections. The architects Eero Saarinen and Hannskari Bandal de- signed the St. Louis Gateway Arch by using the hyperbolc cosine function. Indeed the inside shape of the Arch is a slightly flattened catenary. You have probably observed catenary curves. That is the shape of a perfictly flexible ca- ble hanging between two posts, and the chains that often hang on the borders of gardens to prevent people from walking on grass are approximate catenaries. More importantly, engineers use these functions in various design problems, for example in the design of high voltage lines. It is interesting to note that, from Euler's relation we get the two complex exponential formulae ei x − e−i x ei x + e−i x sin (x) = ; and cos(x) = : 2i 2 This, together with the definitions of the hyperbolic sine and cosine suggests that there are formulae that relate the trigonometric to the hyyperbolic functions and this is indeed the case.
[Show full text]
Hyperbolic Functions∗
Hyperbolic functions∗ Attila Máté Brooklyn College of the City University of New York November 30, 2020 Contents 1 Hyperbolic functions 1 2 Connection with the unit hyperbola 2 2.1 Geometricmeaningoftheparameter . ......... 3 3 Complex exponentiation: connection with trigonometric functions 3 3.1 Trigonometric representation of complex numbers . ............... 4 3.2 Extending the exponential function to complex numbers. ................ 5 1 Hyperbolic functions The hyperbolic functions are defined as follows ex e−x ex + e−x sinh x ex e−x sinh x = − , cosh x = , tanh x = = − 2 2 cosh x ex + e−x 1 2 sech x = = , ... cosh x ex + e−x The following formulas are easy to establish: (1) cosh2 x sinh2 x = 1, sech2 x = 1 tanh2 x, − − sinh(x + y) = sinh x cosh y + cosh x sinh y cosh(x + y) = cosh x cosh y + sinh x sinh y; the analogy with trigonometry is clear. The differentiation formulas also show a lot of similarity: ′ ′ (sinh x) = cosh x, (cosh x) = sinh x, ′ ′ (tanh x) = sech2 x = 1 tanh2 x, (sech x) = tanh x sech x. − − Hyperbolic functions can be used instead of trigonometric substitutions to evaluate integrals with quadratic expressions under the square root. For example, to evaluate the integral √x2 1 − dx Z x2 ∗Written for the course Mathematics 1206 at Brooklyn College of CUNY. 1 for x > 0, we can use the substitution x = cosh t with t 0.1 We have √x2 1 = sinh t and dt = sinh xdx, and so ≥ − √x2 1 sinh t − dx = sinh tdt = tanh2 tdt = (1 sech2 t) dt = t tanh t + C.
[Show full text]
Hyperbolic Functions
HYPERBOLIC FUNCTIONS The following worksheet is a self-study method for you to learn about the hyperbolic functions, which are algebraically similar to, yet subtly different from, trigonometric functions. In order to complete the worksheet, you need to refer back to topics from trigonometry, precalculus and differential calculus. In particular, you may need to refresh yourself on the following: Trigonometry: quotient, reciprocal, Pythagorean, sum/difference of angles, double angle identities (you only need to know the identities, not how to prove them) Precalculus: determining if a function has odd/even symmetry algebraically and graphically determining if a function is one-to-one graphically relationship between the graph/domain/range of a function and its inverse using function composition to determine if two functions are inverses of each other solving for the inverse of a function algebraically Calculus: finding vertical/horizontal asymptotes of a function algebraically using L’Hopital’s Rule for appropriate limits proofs of the derivatives of inverse trigonometric functions The solutions of the questions on the worksheet depend on some skills you should already have, as well as some you will develop this quarter. You should already be able to handle all questions except #14 and parts of #8. You will be able to handle those two remaining questions after we cover the chapter on advanced integration techniques. Skills Required Questions On Worksheet Algebra 1, 2, 3a-d, 4, 9, 10, 11 Limits & Differentiation 3e, 5, 6, 8ac, 12, 13 Integration 7, 8bde, 14 This worksheet must be completed before Thanksgiving, and the material on it will appear on the final exam.
[Show full text]
Hyperbolic Functions and Solutions to Second Order Odes
Hyperbolic Functions and Solutions to Second Order ODEs R. C. Daileda March 1, 2018 1 Hyperbolic Functions For any x, the hyperbolic cosine and hyperbolic sine of x are deﬁned to be ex + e−x cosh x = ; 2 ex − e−x sinh x = ; 2 respectively.1 It is straightforward to check that they satisfy the identity cosh2 x − sinh2 x = 1 as well as the derivative formulae d cosh x = sinh x; dx d sinh x = cosh x: dx The names for these functions arise from the fact that they parametrize the (right branch) H of the hyperbola x2 − y2 = 1 in the same manner that the circular functions sine and cosine parametrize the circle x2 + y2 = 1. Namely, if we draw a ray R from the origin into either the ﬁrst or fourth quadrants, and let A denote the area trapped by R, H and the x-axis (A is taken to be negative if R has negative slope), as shown in the diagram below, then the coordinates of the intersection of R and H can be shown to be x = cosh 2A; y = sinh 2A: If we perform the same construction using the unit circle C instead, which is given by x2 + y2 = 1, we obtain x = cos 2A; y = sin 2A: So the analogy between the circular and hyperbolic functions is that they parametrize dif- ferent curves, but in the exact same way. 1These are typically read as \kosh of x" and \cinch of x." 1 The parametrization of the hyper- The parametrization of the unit cir- bola H by the area A, resulting in cle C by the area A, resulting in the the hyperbolic functions.
[Show full text]
Hyperbolic Functions
Chapter 3. Section 6 Page 1 of 5 Section 3.6 – Hyperbolic Functions Definitions: • Recall the trig functions are related to the unit circle xy22+ =1… • In a similar way, hyperbolic functions are related to the hyperbolic function xy22−=1… • They can be expressed in terms of linear combinations of exponential growth and decay. eexx− − • sinh x = 2 eexx+ − cosh x = 2 • For the regular trig functions, sine and cosine give rise to tangent, cotangent, secant and cosecant. • In a similar way, hyperbolic sine and hyperbolic cosine give rise to… sinhx 1 1 cosh x tanhxxxx==== csch sech coth coshx sinhxxx cosh sinh C. Bellomo, revised 26-Oct-09 Chapter 3. Section 6 Page 2 of 5 Graphs of Hyperbolic Sine and Cosine: eexx− − • sinh x = 2 eexx+ − • cosh x = 2 C. Bellomo, revised 26-Oct-09 Chapter 3. Section 6 Page 3 of 5 Properties of Hyperbolic Functions: ee−−−xx−−() ee xx − • sinh(−x ) = =− =− sinh x 22 ee−−−xxxx++() ee − • cosh(−=x ) = = cosh x 22 • Q: From our definitions of even and odd, how can we classify sinh and cosh? A: Sinh is an odd function, cosh is an even function. • Q: What can we then say about their symmetry? A: Sinh is symmetric about the origin, cosh is symmetric about the y axis. • cosh22x −= sinhx 1 • And the last two properties are: sinh(x +=yxyxy ) sinh cosh + cosh sinh cosh(x +=yxyxy ) cosh cosh + sinh sinh Derivatives: ddeeee⎛⎞xx−+−− xx • sinhx ===⎜⎟ cosh x . dx dx ⎝⎠22 The derivative of sinh is cosh ddeeee⎛⎞xx+−−− xx • coshx ===⎜⎟ sinh x .
[Show full text]