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Is there a solution to this transcendental equation ?

  1. Sep 29, 2014 #1
    This is not a homework problem, this equation simply occurred to my mind and my math teacher said such an equation either can't exist or he doesn't know the answer.
    sinX ➕ cosX = lnX

    I don't know how to start....
     
  2. jcsd
  3. Sep 29, 2014 #2
    This looks ugly. To simplify it a bit, notice that the LHS is equal to ##\frac12 \sin (2x)##. In particular, this lies in ##\left[-\frac12,\frac12\right]##, so that any ##x## solving your equation would have to belong to ##\left[\frac1{\sqrt{e}},\sqrt{e}\right]##.

    Using some knowlege of the functions ##\sin## and ##\ln## (i.e. knowing where each of them is positive/negative and where each of them is positively/negatively sloped, and using that ##2\in (0,\pi)##), you can use intermediate value theorem to show that the equation has a unique solution ##x^*>0##, and that it satifies ##x^*\in (1, \sqrt{e})##.

    As for an explicit solution, good luck.
     
  4. Sep 29, 2014 #3

    PeroK

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    You could have started by drawing a graph of each function. Note that:

    ##sin(x) + cos(x) = \sqrt{2}sin(x + \frac{\pi}{4})##

    Which simplifies things.
     
  5. Sep 29, 2014 #4

    PeroK

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    ##sin(x) + cos(x) \ne \frac12 \sin (2x)##

    ##sin(x)cos(x) = \frac12 \sin (2x)##
     
  6. Sep 29, 2014 #5
    Was it "+" rather than "*" ?

    On my computer, it's showing up as "sin(x) [black square with a white X in it] cos(x)". If it was addition, my bad. Sorry for adding confusion.
     
  7. Sep 29, 2014 #6

    mfb

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    I think it is some special character, something like a large "+" sign. So it is clearly an addition.
    This does not change the result - there is a unique solution, but probably no closed form for it.
     
  8. Sep 30, 2014 #7
    with intermediate value method X is something like 1.8893....
     
  9. Sep 30, 2014 #8

    Char. Limit

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    Indeed. Which is the sole numerical solution to sin(x)+cos(x) = ln(x).

    It has no closed form, though. Numerical solution is the best you're going to get.
     
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