The Hyperbolic Function
Output of MD2002 Program ABH
Question and Answer
Questions


  • Q01 | - Hyperbolic functions
  • Q02 | - Graphs of hyperbolic functions
  • Q03 | - Compare y=sin(x) and y=sinh(x)
  • Q04 | - The curve of y=sin(x) and y=sinh(x) are different. Why name sinh(x) ?
  • Q05 | - Prove taht sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)
  • Q06 | - Prove that sinh(x)^2 - cosh(x)^2 = -1
  • Q07 | - If y=sinh(x) then y'=cosh(x)
  • Q08 | - If y = cosh(x) then y' = sinh(x)
  • Q09 | - Prove that sinh(2*x) = 2*sinh(x)*cosh(x)
  • Q10 | -
    • Answers


    Q01. Hyperbolic functions

    Definitions
      1. sinh(x) = (exp(x)-exp(-x))/2
      2. cosh(x) = (exp(x)+exp(-x))/2
      3. tanh(x) = sinh(x)/cosh(x)
      4. csch(x) = 1/sinh(x)
      5. sech(x) = 1/cosh(x)
      6. coth(x) = 1/tanh(x)

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    Q02. Graphs of hyperbolic functions

  • Diagrams
    Click start and select subject
    • y=sinh(x) .... Program 06 01
    • y=cosh(x) .... Program 06 02
    • y=tanh(x) .... Program 06 03
    • y=csch(x) .... Program 06 04
    • y=sech(x) .... Program 06 05
    • y=coth(x) .... Program 06 06
  • See also PM section 14
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    Q03. Compare y=sin(x) and y=sinh(x)

  • Diagram
    Click start and select subject
    • y=sin(x) ........ Program 05 01
    • y=sinh(x) ....... Program 06 01
  • See also PM Section 14
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    Q04. The curve of y=sin(x) and y=sinh(x) are different. Why name sinh(x) ?

    Answer based similar identities
      1. Sine function
      • sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y)
      • sin(x-y) = sin(x)*cos(y) - cos(x)*sin(y)
      2. Hyperbolic Sine function
      • sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)
      • sinh(x-y) = sinh(x)*cosh(y) - cosh(x)*sinh(y)
    Answer based similar identities
    • sin(x)^2 + cos(x)^2 = 1 is unit circle
    • sinh(x)^2 - cosh(x)^2 = -1 is unit hyperbola
    Reference : See PM section 14
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    Q05. Prove taht sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)

    Proof : Use exponential formula
    • lhs = sinh(x+y) = (exp(x+y)-exp(-x-y))/2
    • = [exp(x)*exp(y) - exp(-x)*exp(-y)]/2
    • rhs = (exp(x)-exp(-x))*(exp(y)+exp(-y))/4
    • + (exp(x)+exp(-x))*(exp(y)-exp(-y))/4
    • = [exp(x)*exp(y) + exp(x)*exp(-y) - exp(-x)*exp(y) - exp(-x)*exp(-y)]/4
    • + [exp(x)*exp(y) - exp(x)*exp(-y) + exp(y)*exp(-x) - exp(-x)*exp(-y)]/4
    • = [exp(x)*exp(y) - exp(-x)*exp(y)]/2
    • lhs = rhs

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    Q06. Prove that sinh(x)^2 - cosh(x)^2 = -1

    Proof : Use exponential formula
    • lhs = (exp(x)-exp(-x))/2)^2 - (exp(x)+exp(-x))/2)^2
    • = [exp(2*x) - 2 + exp(-2*x)]/4 - [exp(2*x) + 2 +exp(-2*x)]/4
    • = -1 = rhs
    Locus of x = sinh(t) and y = cosh(t)
    • x^2 - y^2 = sinh(t)^2 - cosh(t)^2 = -1.
    • Hence the locus is unit hyperbola.

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    Q07. If y=sinh(x) then y'=cosh(x)

    Proof
    • y = sinh(x) = (exp(x)-exp(-x))/2
    • y'= (exp(x)+exp(-x))/2 = cosh(x)
    Note
    • if y=sin(x) and then y'=cos(x)
    • This another reason we name the function as sinh(x)

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    Q08. If y=cosh(x) then y'=sinh(x)

    Proof
    • y = cosh(x) = (exp(x)+exp(-x))/2
    • y'= (exp(x)-exp(-x))/2 = sinh(x)
    Note
    • if y=cos(x) and then y'=sin(x)
    • This another reason we name the function as cosh(x)

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    Q09. Prove that sin(2*x) = 2*sinh(x)*cosh(x)

    Proof : Using sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)
    • Let x = y
    • Hence sinh(2*x) = 2*sinh(x)*cosh(y)
    Proof : Using exponent law
    • sinh(x) = (e^x - e^(-x))/2
    • cosh(x) = (e^x + e^(-x))/2
    • Hence 2*sinh(x)*cosh(x)
    • = 2*(e^x - e^(-x))*(e^x + e^(-x))/4
    • = (e^(2*x) - e^(-2*x))/2
    • = sinh(2*x)

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    Q10. Answer
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