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Trigonometric Equations

  1. May 22, 2007 #1
    How would one go about solving sin(x) = x/2
    I.e. the intersections of

    I can rigorously solve this by going to each individual period and finding the intersections. But is there a better way?
  2. jcsd
  3. May 22, 2007 #2
    You can find the no. of solutions easily enough using graphs, but getting the actual solution, would require rigour.
  4. May 22, 2007 #3
    This is going to do my scholarship exam on Friday. I was only notified today that I was selected. Although I knew most of the requirements prior to this, I did not know about trigonometric equations such as this. I cannot afford to draw graphs - time constraints are not going to do me favours. I know for sin(x)=k, there is a simple general solution rule. But as for kx; are there any quicker methods?
  5. May 23, 2007 #4
    Do you want to know the exact intersections or the number of intersections? If it's the number, it's fairly easy. The maxima and minima of sin x all have y = 1 and y = -1. The function x/2 is equal to 1 at x = 2 and - 1 at x = -2. Since the pi/2< 2 <pi, then it has to cross sin x at two points on the positive x axis (picture this in your mind: the line has to cross the "mountain" between 0 and pi). Same applies to -2 > -pi. There are in total 3 intersection points (x = 0 is common to the positive and negative sides of the x axis).
  6. May 23, 2007 #5
    Yes I suppose knowing the number of solutions may be helpful. However, the values are also expected...
  7. May 23, 2007 #6
    Then, there exist no analytical method. Apart from the obvious x = 0 solution, the others have to be found by other method. Are you familiar with Newton's method?
  8. May 23, 2007 #7
    No I am not aware of Newton's method
  9. May 23, 2007 #8
    Then learn about it, you haven't got much time! Though, it's strange that they would ask you this kind of question...
    Last edited: May 23, 2007
  10. May 23, 2007 #9
    Wow! I have just learnt it! It is quite accurate with just 3 steps. Since they expect 3 s.f. it is perfect. Thanks.
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