I recently started studying simple harmonic motion and I came across two equations for the displacement of a particle, as given in my textbook:

\begin{align} y &= a\sin(\omega t +\phi)\\ x &= a\cos(\omega t+ \phi) \, . \end{align}

I really have no clue when to use which one. It would help me a lot if someone could give an explanation on this.


1 Answer 1


They're equivalent. You can always use either one. To convert between the two, use the basic trigonometric identity

$$\cos(\omega t+\phi)=\sin(\omega t+\phi-\pi/2)$$

  • $\begingroup$ I see, and will the signs of ๐“ change depending on whether the motion is clockwise or anticlockwise? $\endgroup$
    – user236322
    Jan 10, 2020 at 18:38
  • 2
    $\begingroup$ @UserKunal123 No, the sign of $\omega$ will change. $\phi$ defines the starting point of the oscillation. $\endgroup$ Jan 10, 2020 at 18:40
  • $\begingroup$ I get it..but there is one more equation given in my book: y=asin(wt-๐“). Isn't the sign of ๐“ changing here? And when do I use this equation? I apologize for not being clear. $\endgroup$
    – user236322
    Jan 10, 2020 at 18:53
  • $\begingroup$ For the equation $y=A\sin(\omega t-\phi)$, the starting position is positive if $\phi$ is positive. For the equation $y=A\sin(\omega t+\phi)$, the starting position is negative if $\phi$ is positive. Once again, you can use either one. $\endgroup$ Jan 10, 2020 at 18:55
  • 1
    $\begingroup$ @UserKunal123 Yes, the numbers you plug in to $\omega$ and $\phi$ would be different in every case, but you can use any of them. $\endgroup$ Jan 10, 2020 at 19:10

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