> Period is independent of amplitude. ([Vias.org][1])

But given that,
> Simple harmonic motion can be defined by 

>1) $$x = A * sin  (ωt)$$ 
where A= amplitude, w= angular velocity, t=time, x=displacement from the mean position

And,

>2)$$T - 2 π/ω = 1/f$$ 
where, T= period of motion, f=frequency

1) can be rearranged to give
$$x = A * sin  (ωt)$$ 
$$x/A = sin  (ωt)$$ 
$$arcsin(x/A) = ωt$$ 
$$ω=arcsin(x/A)/t$$   

Subbing this into 2) gives the following relationship between $T$ and $A$


$$T - 2 π/(arcsin(x/A)/t)   = 1/f$$

**Doesn't the fact that both $T$ and $A$ appear in the above equation show that T(period) is dependent on A(amplitude) ?** 

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FYI:

Although a physical explanation may be useful, **I am particularity interested in why deriving a relationship between $T$ and $A$ doesn't mean that they are dependent**. Note there is a similar question [here][4] but that is concerned with the physics of the phenomena, and **not** why the maths can't be used to solve it.

This is because I have this exam where we are given a stimulus and based on that stimulus alone are meant to answer questions (i.e. the exam is expected to contain material/principles that we haven't been exposed to before, but should be able to answer given the stimulus). And as I didn't know much about simple harmonic motion, my initial reaction was to see if the formulas link $T$ and $A$.

The practice question (which I have typed out the important bits of above is)
[![Q][2]][2]
[![Q][3]][3]
The answer is D.


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  [1]: http://www.vias.org/physics/bk3_01_03.html
  [2]: https://i.sstatic.net/6GDZC.png
  [3]: https://i.sstatic.net/K2R13.png
  [4]: http://physics.stackexchange.com/questions/259773/independence-of-period-and-amplitude-in-simple-harmonic-motion