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File: Calculus Pdf 169524 | Che205s17 Reading 08b
hyperbolic functions cheatsheet 1 intro for historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus course but as a matter of fact ...

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                                    Hyperbolic functions (CheatSheet)
              1    Intro
              For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus
              course, but as a matter of fact they have the same dignity as trigonometric functions. Unfortu-
              nately this can be completely understood only if you have some knowledge of the complex numbers.
              Roughly speaking ordinary trigonometric functions are trigonometric functions of purely real num-
              bers, and hyperbolic functions are trigonometric functions of purely imaginary numbers. For the
              moment we have to postpone this discussion to the end of Calc3 or Calc4, but still we should be
              aware of the fact that the impressive similarity between trig formulas and hyperbolic formulas is
              not a pure coincidence.
                 Most of the formulas that follow correspond precisely to a trig formula or they differ by at most
              a change of sign. For each formula I will explicitly state if some change of sign occurs or not (the
              different sign is marked in green).
                 The main purpose of this paper is not to give you a bunch of formulas to memorize, but to
              make you aware of the fact that hyperbolic formulas are just like trig formulas up to signs; and
              correct signs can always be checked with some very quick calculation.
              2    Definitions
              Definition of hyperbolic sine and cosine:
                                                            x   −x
                                                  sinhx = e −e
                                                              2
                                                            x   −x
                                                  coshx = e +e
                                                              2
              There are two equivalent formulas for sine and cosine (Euler’s formulas) but they require some
              knowledge of the complex numbers:
                                                           ix   −ix
                                                   sinx = e  −e
                                                             2i
                                                           ix   −ix
                                                   cosx = e +e
                                                              2
                        √                  2
              where i =  −1 or if you prefer i = −1. Substituting x with ix in these two formulas and keeping
              in mind that i2 = −1 it’s immediate to deduce that coshx = cos(ix) and sinhx = −isin(ix)
              (I mention this just for the sake of completeness and because it’s fun!). From the definition of
              hyperbolic sine and cosine we define hyperbolic tangent, cotangent, secant, cosecant in the same
                                                          1
             way we did for trig functions:
                                                tanhx = sinhx
                                                         coshx
                                                cothx = coshx
                                                        sinhx
                                                sechx =   1
                                                        coshx
                                                cschx =   1
                                                        sinhx
             3    Basic properties
             First of all we notice that hyperbolic functions have the same parity as the corresponding trig
             functions:
                                              sinh(−x) = −sinhx                               (1)
                                              cosh(−x) = coshx                                (2)
                                              tanh(−x) = −tanhx                               (3)
                                              coth(−x) = −cothx                               (4)
                                              sech(−x) = sechx                                (5)
                                              csch(−x) = −cschx                               (6)
                                                                                              (7)
             All these formulas follow immediately from the definitions we gave in the previous section. Unlike
             trig functions, hyperbolic functions are not periodic!
                Using the definition of hyperbolic sine and cosine it’s possible to derive identities similar to
                2      2            2          2
             cos x+sin x=1 and tan x+1=sec x:
                                              cosh2x−sinh2x = 1                               (8)
                                                  2      2
                                              tanh x+sech x = +1                              (9)
             These identities do not require Pythagoras’ theorem, they can be derived from the definition with
             a direct calculation and using properties of the exponential.
             4    Addition formulas
             Unlike the case of ordinary addition trig formulas, the two basic addition formulas for hyperbolic
             functions can be retrieved immediately from the definition.
                                      sinh(x+y)=sinhxcoshy+coshxsinhy                        (10)
                                      cosh(x+y)=coshxcoshy+sinhxsinhy                        (11)
             Combining these formulas with (1) we easily derive the following:
                                      sinh(x−y)=sinhxcoshy−coshxsinhy                        (12)
                                      cosh(x−y)=coshxcoshy−sinhxsinhy                        (13)
                                                       2
                Using the definition of hyperbolic tangent and equations (10,11) we can derive the addition formula:
                                                   tanh(x±y)= tanhx±tanhy                                         (14)
                                                                   1±tanhxtanhy
                5     Formulas for the double and half angle
                Using equations (10,11) with x = y we immediately have:
                                                      sinh2x = 2sinhxcoshx                                        (15)
                                                                     2         2
                                                      cosh2x = cosh x+sinh x                                      (16)
                By plugging (8) into (16) we have the following two formulas for the squares of sine and cosine:
                                                        cosh2x = 1+cosh2x                                         (17)
                                                                        2
                                                             2     cosh2x−1
                                                        sinh x =        2                                         (18)
                Notice that both (16) and (8) differ from the corresponding trig formulas by a sign, but the resulting
                                 2                                                                      2
                formula for cosh is the same as in the trigonometric case, and the formula for sinh has a global
                change of sign. By substituting x with x and taking the square root we have formulas for the half
                                                          2
                angle:
                                                       cosh x = r1+coshx                                          (19)
                                                             2     r 2
                                                       sinh x = ±    coshx−1                                      (20)
                                                            2             2
                                               1
                The first formula is the same as the trigonometric one, and in the second one we have a global
                change of sign in the radicand. In the same way, but using (14) we have:
                                                        tanh2x = 2tanhx                                           (21)
                                                                            2
                                                                    1+tanh x
                6     Inverse Hyperbolic functions
                It’s easy to check that hyperbolic sine is a monotonic increasing function on the real numbers, and
                for this reason it’s invertible on all the real axis . Hyperbolic cosine has a global minimum at x = 0
                whose value is y = 1 and it’s decreasing from −∞ to 0 and increasing from 0 to +∞, for this reason
                we can invert it on the positive half-axis or the negative one. By convention we choose the positive
                one [0,∞). Hyperbolic tangent is defined for all real numbers, it’s monotonic increasing and it has
                horizontal asymptotes:
                                                           lim tanhx = ∓1                                         (22)
                                                          x→∓∞
                   1You don’t need to choose the sign in front of the radical since cosh is always positive
                                                                   3
                The inverse of an hyperbolic function can always be written as the logarithm of an algebraic2
                function:
                                                  p 2
                              arsinhx = ln(x+       x +1),     Domain=(−∞,+∞), Range=(−∞,+∞)                       (23)
                                                  p 2
                              arcoshx = ln(x+       x −1),     Domain=[1,+∞), Range=[0,+∞)                         (24)
                                           1   1+x
                              artanhx = 2 ln 1−x ,         Domain=(−1,1), Range=(−∞,+∞)                            (25)
                Rememberthat the domain of the inverse is the range of the original function, and the range of the
                inverse is the domain of the original function. To retrieve these formulas we rewrite the definition
                                                                               x                      x
                of the hyperbolic function as a degree two polynomial in e ; then we solve for e and invert the
                exponential. For example:
                                               x     −x                                        p
                                              e −e            2x      x               x            2
                                y = sinhx =       2      ⇔ e −2ye −1=0 ⇔ e =y± y +1
                and since the exponential must be positive we select the positive sign.
                7     Derivatives
                The calculation of the derivative of an hyperbolic function is completely straightforward, so I will
                just report a list of formulas with no additional comments:
                                                       d sinhx = coshx                                             (26)
                                                      dx
                                                       d coshx = +sinhx                                            (27)
                                                      dx
                                                       d tanhx = sech2x                                            (28)
                                                      dx
                                                       d cothx = −csch2x                                           (29)
                                                      dx
                                                       d sechx = −tanhxsechx                                       (30)
                                                      dx
                                                       d cschx = −cothxcschx                                       (31)
                                                      dx
                Formulas for the derivative of an inverse hyperbolic function can be quickly calculated from (23)
                using basic properties of derivatives. They can also be calculated using the formula for the derivative
                of the inverse:
                                                        d arsinhx = √      1                                       (32)
                                                       dx                   2
                                                                        +x +1
                                                        d arcoshx = √ +1                                           (33)
                                                       dx                 2
                                                                         x −1
                                                        d artanhx =      1                                         (34)
                                                                           2
                                                       dx              1−x
                   2An algebraic function is a function containing the four operations and radicals only.
                                                                   4
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