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Solving a simple equation

  1. Apr 25, 2004 #1
    I am trying to solve this equation:

    [tex] Tan[\gamma a]=-\frac{\gamma}{\beta} [/tex]

    where [itex] \beta [/itex] and a are just numbers and I am trying to solve for [itex] \gamma [/itex]. I tried graphing it but I don't see how the solution varies with the choice of a, which is a free parameter in the problem.
  2. jcsd
  3. Apr 26, 2004 #2
    You can solve by drawing the graph of both Tan[xa] and -x/b and seeing when they intersect. (i used x in place of the symbol you gave)

    thus it has many solutions as tan is a trigonometric function.

    If i took a as 4 and b as 5 then
    x can equal 0, Tan[0] = 0
    x can equal 3, Tan[12] = -3/5 (approximatly)

    this is from using a graphics calculator and finding where the two graphs intersect.

    There are infinate more values.

    i do not know of any other way to solve for the unknown. Yet.
  4. Apr 26, 2004 #3
    Yes I understand how to do that... maybe I didn't phrase my question well. By holding [itex] \beta [/itex] constant and varying a, I can obtain and bunch of numbers for [itex] \gamma [/itex] and then fit the curve to obtain how [itex] \gamma [/itex] varies. What I was wondering was if there was another way to do this, that doesn't force me to use curve fitting. Any ideas anyone? Or maybe an easy way to see how [itex] \gamma [/itex] varies when changing a.
  5. Apr 26, 2004 #4
    [tex]\tan {\gamma a} = -\frac{\gamma}{\beta}[/tex]

    Treat a as a function of gamma and differentiate.

    [tex]\sec^2{\gamma a} \Big( a + \gamma \frac{da}{d\gamma} \Big) = -\frac{1}{\beta}[/tex]

    Solve for [itex]d\gamma[/itex].

    [tex]d\gamma = \frac{-a \beta da}{\cos^2{\gamma a} + a\beta}[/tex]

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