# Which equation to use for simple harmonic motion?

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

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

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

• I see, and will the signs of 𝝓 change depending on whether the motion is clockwise or anticlockwise? – user236322 Jan 10 '20 at 18:38
• @UserKunal123 No, the sign of $\omega$ will change. $\phi$ defines the starting point of the oscillation. – probably_someone Jan 10 '20 at 18:40
• I get it..but there is one more equation given in my book: y=asin(wt-𝝓). Isn't the sign of 𝝓 changing here? And when do I use this equation? I apologize for not being clear. – user236322 Jan 10 '20 at 18:53
• 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. – probably_someone Jan 10 '20 at 18:55
• @UserKunal123 Yes, the numbers you plug in to $\omega$ and $\phi$ would be different in every case, but you can use any of them. – probably_someone Jan 10 '20 at 19:10