# How can we use sin/cos for things like simple harmonic motion or any periodic type?

The sine and cosine functions are both well defined periodic function with fixed period and amplitude. My understanding is that for phenomena that are periodic, since sine and cosine are well defined are being utilized as a way to come up with a formula that describes them thus treating sine and cosine as black box functions.

In order to use them we need to change the scale since sine and cosine are between $-1$ and $1$ for amplitude and $x$ and $x + 2\pi$ for the period. An example usage would be if we change scale of $y = \cos x$ to be $y = A\cos(\pi x/P)$ and then replace to create a function of time: $$f(t) = A\cos(2\pi t/Τ)\;.$$

So I am confused on the following: sin and cos are both defined in terms of radians (I mean the period they have is $2\pi$). How can we get meaningful results then to describe other things besides degrees? It seems to me we are just passing any value to sine and cosine and ignore that they are defined in terms of radians. Can someone please help me clarify this?

• You can think of it as a conversion from whatever units $t$ has (let's say time) to radians. If it takes time $T$ to move $2\pi$ radians, then at time $t$ we are at $2\pi t/T$ radians. Then it makes sense to take the cosine of that value. – Rahul Aug 28 '16 at 16:18
• @Rahul:But in periodic motion the displacement is not in 2π terms right? – Jim Aug 28 '16 at 16:52
• Radian is to be considered a name, indication, not a dimension unit. the radian is the ratio of an arc by a radius, so it is dimensionless, same as it is $\pi$, and same as is $t/T$. – G Cab Sep 4 '16 at 21:11