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Mathematics Dictionary
Dr. K. G. Shih
Hyperbolic Functions
Subjects


  • AN 04 01 | - Definition
  • AN 04 02 | - Graph of y = sinh(x)
  • AN 04 03 | - Relation between functions and its inverse
  • AN 04 04 | - Comparison with trigonometric functions
  • AN 04 05 | - Why do we name the function as sinh(x) ?
  • AN 04 06 | - cosh(x) and cos(x) : comparison
  • AN 04 07 | -
  • AN 04 08 | -
  • AN 04 09 | -
  • AN 04 10 | -
    • Answers


    AN 04 01. Defintions

    Reference     Definition of hyperbolic functions
    • sinh(x) = (exp(x) - exp(-x))/2
    • cosh(x) = (exp(x) + exp(-x))/2
    • tanh(x) = sinh(x)/tanh(x)
    • csch(x) = 1/sinh(x)
    • sech(x) = 1/cosh(x)
    • coth(x) = cosh(x)/sinh(x)
    Definition of inverse hyperbolic functions
    • arcsinh(x) = ln(x + Sqr(x^2 + 1))
    • arccosh(x) = ln(x + Sqr(x^2 - 1))
    • arctanh(x) = ln((1 + x)/(1 - x))
    • arccsch(x) = ln(1/x + Sqr(1/x^2 + 1))
    • arcsech(x) = ln(1/x + Sqr(1/x^2 - 1))
    • arccoth(x) = ln((1 - x)/(1 + x))

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    AN 04 02. Diagrams

    • Study Subject Diagrams of hyperbolic functions
    • Functions
      • Graph of y = sinh(x)
      • Graph of y = cosh(x)
      • Graph of y = tanh(x)
      • Graph of y = csch(x)
      • Graph of y = sech(x)
      • Graph of y = coth(x)

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    AN 04 03. Relation between function and its inverse
    Relation 1
    • Sinh(arcsinh(x)) = x
    • Cosh(arccosh(x)) = x
    • Tanh(arctanh(x)) = x
    Relation 2
    • Arcsinh(sinh(x)) = x
    • Arccosh(cosh(x)) = x
    • arctanh(tanh(x)) = x
    Reference : Examples to prove
    • Picture Mathematics by Dr. Shih. P129 - P135

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    AN 04 04. Comparison with trigonometric functions

    Diagrams     Texts Example : From diagram, describe the similarity and difference of functions
    • 1. y = sin(x) and y = sinh(x)
    • 2. y = cos(x) and y = cosh(x)
    • 3. y = tan(x) and y = tanh(x)
    • 4. y = csc(x) and y = csch(x)
    • 5. y = sec(x) and y = sech(x)
    • 6. y = cot(x) and y = coth(x)

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    AN 04 05. Why do we name the function (e^x - e^(-x))/2 as sinh(x) ?

    Text : Similarity of sin(x) and sinh(x) by identities     Describe the curves of y = sin(x) and y = sinh(x)
    • The curve of y = sin(x)
      • It is periodic function with period of 2*pi
      • It range for y = -1 to y = 1
      • It has zero value at x = 0, pi, 2*pi, ....
      • From x = -pi/2 to 0
        • The curve is incresing (y' is positive)
        • The curve is concave upward (y" is positive)
      • From x = 0 to pi/2
        • The curve is incresing (y' is positive)
        • The curve is concave downward (y" is negative)
      • For x = pi/2 to pi
        • The curve is decresing (y' is negative)
        • The curve is concave downward (y" is negative)
      • ....
    • The curve of y = sinh(x)
      • The range is from -infinite to +infinite
      • It has zero value at x = 0, pi, 2*pi, ....
      • From x = 0 to infinite
        • The curve is incresing (y' is positive)
        • The curve is concave upward (y" is positive)
      • From x = -infinite to 0
        • The curve is incresing (y' is positive)
        • The curve is concave downward (y" is negative)
    • Conclusion : form x = -pi/2 to pi/2
      • Both has zero value at x = 0
      • Both curves are increasing but different concavity
      • The range are completely different

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    AN 04 06. cosh(x) and cos(x) : comparison
    Text : Similarity of cos(x) and cosh(x) by identities     Describe the curves of y = cos(x) and y = cosh(x)
    • The curve of y = cos(x)
      • It is periodic function with period of 2*pi
      • It range for y = -1 to y = 1
      • It has zero value at x = pi/2, 3*pi/2, ....
      • From x = -pi/2 to 0
        • The curve is incresing (y' is positive)
        • The curve is concave downward (y" is negative)
      • From x = 0 to pi/2
        • The curve is decresing (y' is negative)
        • The curve is concave downward (y" is negative)
      • For x = pi/2 to pi
        • The curve is decresing (y' is negative)
        • The curve is concave upward (y" is positive)
      • ....
    • The curve of y = cosh(x)
      • The range is from -infinite to 1 at x = 0 and then to +infinite
      • It has no zero value and the values are always positive
      • From x = -infinite to 0
        • The curve is decresing (y' is negative)
        • The curve is concave upward (y" is positive)
      • From x = 0 to +infinite
        • The curve is incresing (y' is positive)
        • The curve is concave upward (y" is positive)
    • Conclusion : form x = -pi/2 to pi/2
      • Both has value 1 at x = 0
      • Cos(x) is increasing and cosh(x) is decreasing when x between -pi/2 and 0
      • Cos(x) is decreasing and cosh(x) is increasing when x between 0 and pi/2
      • The range are completely different

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    AN 04 07. Answer

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    AN 04 08. Answer

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    AN 04 09. Answer

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    AN 04 10. Answer

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