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Hyperbolic FUNCTIONS

Hyperbolic FUNCTIONS. HYPERBOLIC FUNCTIONS. Certain combinations of the exponential functions e x and e -x arise so frequently in mathematics and its applications that they are given special names. HYPERBOLIC FUNCTIONS.

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Hyperbolic FUNCTIONS

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  1. Hyperbolic FUNCTIONS

  2. HYPERBOLIC FUNCTIONS Certain combinations of the exponential functions exand e-x arise so frequently in mathematics and its applications that they are given special names.

  3. HYPERBOLIC FUNCTIONS In many ways, they are analogous to the trigonometric functions, and they have the same relationship to the hyperbola that the trigonometric functions have to the circle. For this reason, they are collectively called hyperbolic functions and individually called hyperbolic sine, hyperbolic cosine, and so on. Circle x2 + y2 = 4 Hyperbola x2 - y2 = 4

  4. The behavior of these functions shows such remarkable parallels to trig functions, that they have been given similar names. Hyperbolic Sine: (pronounced “shine x”) Hyperbolic Cosine: (pronounced “cosh x”)

  5. DEFINITION

  6. Circle x2 + y2 = 4 Hyperbola x2 - y2 = 4 Parametric coordinate of point on the circle (2cos, 2sin ) Parametric coordinate of point on the hyperbola (2cosh, 2sinh )

  7. HYPERBOLIC FUNCTIONS The graphs of hyperbolic sine and cosine can be sketched using graphical addition, as in these figures.

  8. HYPERBOLIC FUNCTIONS   Note that sinh has domain and Range , whereas cosh has domain and range . 

  9. HYPERBOLIC FUNCTIONS The graph of tanh is shown. It has a horizontal asymptotes y = ±1.

  10. APPLICATIONS The most famous application is the use of hyperbolic cosine to describe the shape of a hanging wire.

  11. APPLICATIONS It can be proved that, if a heavy flexible cable is suspended between two points at the same height, it takes the shape of a curve with equation y = c + a cosh(x/a) called a catenary. The Latin word catena means ‘chain.’

  12. APPLICATIONS Another application occurs in the description of ocean waves. The velocity of a water wave with length L moving across a body of water with depth d is modeled by the function where g is the acceleration due to gravity.

  13. HYPERBOLIC IDENTITIES The hyperbolic functions satisfy a number of identities some of which are similar to well-known trigonometric identities.

  14. HYPERBOLIC IDENTITIES

  15. Osbornes Rule To convert normal trig identities to hyperbolic identities replace cos2x by cosh2x sin2x by – sinh2x This means replace tan2x by – tanh2x as tanh2x contains a sinh2x term and cot2x by –coth2x as coth2x contains a sinh2x term So using cos2x + sin2x = 1 cosh2x – sinh2x = 1 1 + tan2x = sec2x 1 – tanh2x = sech2x 1 + cot2x = cosec2x 1 – coth2x = –cosech2x

  16. INVERSE HYPERBOLIC FUNCTIONS You can see from the figures that sinh and tanh are one-to-one functions. So, they have inverse functions denoted by sinh-1 and tanh-1.

  17. INVERSE FUNCTIONS • This figure shows that cosh is not one-to-one. • However, when restricted to the domain • [0, ∞], it becomes one-to-one.

  18. INVERSE FUNCTIONS The inverse hyperbolic cosine function is defined as the inverse of this restricted function.

  19. INVERSE FUNCTIONS The remaining inverse hyperbolic functions are defined similarly.

  20. INVERSE FUNCTIONS The graph of sinh-1 xis displayed: Reflections in the line y = x.

  21. INVERSE FUNCTIONS The graph of cosh-1xis displayed: Reflections in the line y = x.

  22. INVERSE FUNCTIONS The graph of tanh-1x is displayed: Reflections in the line y = x.

  23. Solving Hyperbolic Equations Use the definitions and hyperbolic identities to solve hyperbolic equations Ex1 Solve sinhx =  Using the definition sinh x = =  2ex – 3 – 2e–x = 0 Multiply by 2ex 2e2x – 3ex – 2 = 0 Quadratic in disguise let ex = z (z – 2)(2z + 1) = 0 z = 2 or – ex = 2 x = ln2 It is impossible for ex = – as ex is always +ve

  24. Ex2 2cosh(2x) + 10 sinh(x) = 5 Using cosh(2x) = 1 + 2sinh2x 2(1 + 2sinh2x) + 10 sinh(x) = 5 4sinh2x + 10sinhx + 2 = 5 4sinh2x + 10sinhx - 3 = 0 Solving this quadratic sinhx = –2.77 or –0.27 x = sinh-1 –2.77 x = –1.743 or x = sinh-1 –0.27 x = –0.2668 Solving Hyperbolic Equations

  25. INVERSE FUNCTIONS Since the hyperbolic functions are defined in terms of exponential functions, it’s not surprising to learn that the inverse hyperbolic functions can be expressed in terms of logarithms.

  26. INVERSE FUNCTIONS In particular, we have: 

  27. INVERSE FUNCTIONS • Show that • See pg 26 in FP2 notes

  28. cannot be ± as this would give the ln of a –ve no.

  29. cannot be ± as this would give the ln of no. between 0 and 1 which is –ve. cosh–1 is always +ve

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