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B Sine/Cosine behaving like a linear function

  1. Jan 11, 2018 #1
    Hello all. After completing a problem in which we derived the formulas for potential and spring force energy as functions of time, with simple harmonic motion I noticed the equations are EXACTLY the same, but with sine and cosine switched. The equations were:

    A sin^2(pi * t)
    A cos^2(pi * t)

    I apologize for this awful format, I want to learn the math script used on this website, but I forgot the name (would be appreciated also).

    Anyways here is a graph of the aforementioned functions. graph of potential energy and spring force energy.png

    My question: Is there a way to represent this back and forth behavior with a linear function or any other way?
  2. jcsd
  3. Jan 11, 2018 #2


    Staff: Mentor

  4. Jan 11, 2018 #3


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    Both functions are not linear.

    If two functions f,g are shifted by a fixed amount with respect to each other, then f(x)=g(x+c) for some constant c. Here ##\cos(x)=\sin(x+\frac \pi 2)##, for example. Sine and cosine are also periodic, so you have additional relations like ##\sin(x)=\sin(x+2n \pi) = \cos(x+2n \pi - \frac \pi 2)## for every integer n.
  5. Jan 12, 2018 #4
  6. Jan 12, 2018 #5


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    This link will help you learn LaTeX for PF
  7. Jan 12, 2018 #6


    Staff: Mentor

    But you can approximate either of these functions in small intervals by straight lines.

    Near x = 0, ##\sin(x) \approx x## and ##\cos(x) \approx 1##. Notice that these are the first terms of the Maclaurin series for each of these functions.
  8. Jan 13, 2018 #7


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    The "back and forth" behavior is a trade-off between the two things. Combining those two into one equation is motivation for Euler's formula involving complex variables: e = cos(θ) + i ⋅ sin(θ). (There is a reason for using θ as though these are functions of angle rather than functions of t=time.) That one equation can be used to represent how the two things trade-off.
    (see https://en.wikipedia.org/wiki/Euler's_formula )

    PS. I see that the wikipedia article goes into a lot of advanced subjects that are not appropriate for this thread. The first paragraph is appropriate. This may be better: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/
  9. Jan 13, 2018 #8

    Charles Link

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    The reason for your graph being what it is in the OP is that ## v=v_o \sin(\omega t) ##, so that the kinetic energy ## K.E.=(\frac{1}{2})mv^2=(\frac{1}{2})mv_o^2 \sin^2(\omega t) ##. In order for the total energy ## E ## to be constant, we must have the potential energy ## U=(\frac{1}{2})m v_o^2 \cos^2(\omega t) ##, because ## \cos^2 (\theta)+\sin^2(\theta)=1 ##. ## \\ ## It may also interest you that ## \cos^2(\omega t)=(\frac{1}{2})(\cos(2 \omega t)+1) ## , and ## \sin^2(\omega t)=(\frac{1}{2})(1-\cos(2 \omega t)) ##. This is why the two curves in the graph of the OP are essentially sinusoidal.
    Last edited: Jan 13, 2018
  10. Jan 23, 2018 #9
    thank you! will learn that asap
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