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Sinh Function

  1. Oct 19, 2010 #1
    1. The problem statement, all variables and given/known data

    sinh(x) = 1

    What is the value of 'x'?


    2. Relevant equations

    sinh(x) = (1/2)(e^x - e^-x)


    3. The attempt at a solution

    e^x - e^-x = 2

    Then what do I do?

    Thanks
     
  2. jcsd
  3. Oct 19, 2010 #2
    Multiply the whole eq by [tex]e^{x}[/tex] then solve the resulting quadtatic eqn in [tex]e^{x}[/tex] and afterwards keep the positive solution and take its logarithm to obtain x.
     
  4. Oct 19, 2010 #3
    That is put, say, [tex]u=e^x[/tex] and solve for u. Then find x from [tex]e^x=u[/tex].
     
  5. Oct 19, 2010 #4
    Thanks. That worked.
     
  6. Oct 19, 2010 #5

    HallsofIvy

    User Avatar
    Science Advisor

    Multiply on both sides by [itex]e^x[/itex] to get [itex](e^x)^2- 1= 2e^x[/itex] and then subtract [itex]2e^x- 1[/itex] from both sides: [itex](e^x)^2- 2e^x= 1[/itex]. Think of that as a quadratic equation in [itex]e^x[/itex] and complete the square: [itex](e^x)^2- 2e^x+ 1= (e^x- 1)^2= 2[/itex].

    Take the square root of both sides, [itex]e^x- 1= \pm\sqrt{2}[/itex] and add 1 to both sides, [itex]e^x= 1\pm\sqrt{2}[/itex]. [itex]1- \sqrt{2}< 0[/itex] so you get the single solution [itex]x= ln(1+ \sqrt{2})[/itex].

    Too slow! Too slow!
     
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