1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    jedishrfu

    Staff: Mentor

  4. Jan 11, 2018 #3

    mfb

    User Avatar
    2017 Award

    Staff: Mentor

    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
    LaTeX.
     
  6. Jan 12, 2018 #5

    lekh2003

    User Avatar
    Gold Member

    https://www.physicsforums.com/help/latexhelp/
    This link will help you learn LaTeX for PF
     
  7. Jan 12, 2018 #6

    Mark44

    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

    FactChecker

    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    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

    User Avatar
    Homework Helper
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Sine/Cosine behaving like a linear function
  1. Sine and cosine (Replies: 4)

  2. Sine and cosine (Replies: 5)

Loading...