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What does (pi/2)+2kpi mean?

  1. Apr 14, 2013 #1
    Hello I was trying to solve this sample problem, but I really dont get it.
    The book says this word by word:

    The equation sin3x=1 implies

    3x=(pi/2)+2kpi, k an integer.
    x= (pi/6)+(2kpi/3), k an integer *Divide each side by 3.

    Because x is not restricted to a finite interval, the given equation has an infinite number of solutions. All the solutions are represented by the equation

    x= (pi/6)+2kpi/3


    Okay it lost me when it told me 3x=(pi/2)+2kpi.
    I am really confused. This is not solved like the rest of the trigonometric equations.
     
  2. jcsd
  3. Apr 14, 2013 #2

    sz0

    User Avatar

    Hi, since the trigonometric functions are periodic there is more than one answer to what 3x can equal to sole your equation.
    sin(v) = 1 implies v = pi/2 , but but pi/2 + 2*pi or pi/2 + 4*pi will do as well since it is 2pi-preiodic. k represents any number, that is k = 0,+-1,+-2,+-3 etc.

    So therefor v = 3x = pi/2 + k*2pi wich gives x= (pi/6)+(2kpi/3)

    if x had been restricted then like for example 0<x<2*pi then only x = pi/6 had been an acceptable answer.
     
  4. Apr 17, 2013 #3

    Hi,

    As the previous poster said, consider the fact that the trig functions are periodic.
    The cosine function can be called a "many-to-one" function.


    So let's think about the function f(x) = cos(x).

    Let me ask you, for what values of x does cos(x) = 1? If you can imagine the graph of the cosine function, or maybe the unit circle, you could tell me: cos(0) = 1.

    But also, cos(-4π) = cos(-2π) = cos(2π) = cos(4π) = 1.

    Therefore, we can say that solutions of x for the equation, are 2π * k, for some integer k.​


    Look at the picture I attached for a view of this.
    https://www.physicsforums.com/attachment.php?attachmentid=57974&stc=1&d=1366232397



    Now, let's apply the same idea to your original question:

    sin(3x) = 1
    What values of (3x) does sin(3x) = 1?
    Well, an obvious one is 3x = π/2, but also 3x = 5π/2, 9π/2 ...

    We can apply the same notation, 3x = π/2 + 2π * k

    Hope this helps.
     

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