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Solving sin(ax-b)

  1. Aug 24, 2015 #1

    adjacent

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    1. The problem statement, all variables and given/known data
    ddd.png
    The figure shows part of a curve with the equation ##y=sin(ax-b)## where ##a>0## and ##0<b<\pi##. The curve cuts the x-axis at the points P, Q and R as shown.
    Given that the coordinates of P, Q and R are
    ##\Big(\frac{\pi}{10},0\Big),\Big(\frac{3\pi}{5},0\Big)## and ##\Big(\frac{11\pi}{10},0\Big)## respectively. Find the values of ##a## and ##b##

    2. Relevant equations


    3. The attempt at a solution
    I know how to solve the this if ##ax-b## was simply ##x##.... ##sin^{-1}(0)## gives 0. So at the interval ##0\leq x \leq 2\pi## ,##x## will be ##\pi-0##(Which is pi)##, 0,2\pi##
    But for this, I did:
    ##\text{let }\alpha = ax-b##
    ##sin^{-1}(0)=0##
    So ##\alpha= 0,\pi,2\pi##
    Looking at the diagram, first point is ##\Big(\frac{\pi}{10},0\Big)## and the value of ##x## is ##\frac{\pi}{10}##. Also the first solution for ##\alpha## is 0. Therefore:
    ##\alpha=0##
    ##a\frac{\pi}{10}-b=0##
    ##a\frac{3\pi}{5}-b=\pi##
    Solving this simultaneously gives ##a=\frac{10}{3}## which is wrong. The actual answer is ##a=2,b=\frac{\pi}{5}## so my method is wrong somehow.
    So what should I do?
     
    Last edited: Aug 24, 2015
  2. jcsd
  3. Aug 24, 2015 #2

    SteamKing

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    Hints:

    1. The difference between R and P should equal the period of the sine function.
    2. The quantity "b" is just a phase angle, which has no effect on the period of the function.
     
  4. Aug 24, 2015 #3

    adjacent

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    But what has this to do with the period of the sine function? Period of ##sin(ax-b)## is ##\pi## . What next?
     
  5. Aug 24, 2015 #4

    SteamKing

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    The period of sin (x) is 2π. What must you do to the argument x to change the period of the sine function from 2π to just π?
     
  6. Aug 24, 2015 #5

    mfb

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    Looks fine up to this point.
    How did you get that answer?
     
  7. Aug 24, 2015 #6

    adjacent

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    Make it 2 :) This method is easy :biggrin:
    Looks like I somehow did my simultaneous equation wrong.

    I got the answer thanks SteamKing and mfb :)
     
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