Rotation and Time periods

Question

I thought I had my head around rotation

$\sin\left(\frac{x}{2}\right)$ has time period $4\pi$,

$\sin(x)$ has time period $2\pi$,

$\sin(2x)$ has time period $\pi$

Yet they all repeat after $2\pi$; so all have a time period that covers $2\pi$ radians on a unit circle but have different time periods above?

Answer 1

Recall the general form of a sine function.

$$y = a\sin[b(x-h)]+k$$

Of course, the only part you’re asking about is $b$, so the rest can be ignored.

$$y = \sin b(x)$$

$b$ shows frequency, or the number of cycles for $0 \leq x \leq 2\pi$.

The period, or wavelength, is given by $\frac{2\pi}{b}$.

$\sin\big(\frac{x}{2}\big)$ will make half a cycle from $0$ to $2\pi$, meaning it has a period of $4\pi$.

$\sin(x)$ will make one full cycle from $0$ to $2\pi$, meaning it has a period of $2\pi$.

$\sin(2x)$ will make two cycles from $0$ to $2\pi$, meaning it has a period of $\pi$.

$\sin\big(\frac{x}{2}\big)$, $\sin(x)$, and $\sin(2x)$ ALL pass through $2\pi$, but the number of cycles in this range of domain will differ. (You seem to have understood the second part, but you’re confused over the first, which is completely irrelevant.)

Maybe checking the graphs could help clarify.

https://www.desmos.com/calculator/nikgf7sfxg

Answer 2

If $x=\pi,$ then $\sin\left(\frac x2\right) = 1.$

Now let’s see if this repeats after $2\pi.$ We can do this by increasing $x$ by $2\pi$ and seeing what the result is. Taking the old value $\pi$ and increasing by $2\pi$ we get $x=3\pi.$

If $x=3\pi,$ then $\sin\left(\frac x2\right) = -1.$ So $\sin\left(\frac x2\right)$ does not repeat after $2\pi.$

On the other hand, if $x=\frac\pi4$ then $\sin(2x)=1.$ But if $x=\frac{5\pi}4$ then $\sin(2x)=1$ again. You can confirm this for any other initial value of $x.$ So $\sin(2x)$ repeats after $\pi.$

That’s the usual interpretation: we regard $\sin\left(\frac x2\right),$ $\sin(x),$ and $\sin(2x)$ as three different functions of $x,$ which have three different minimum periods. You seem to be treating them all as something else, which we might say is $\sin(\theta),$ a single function of $\theta$, where $\theta$ might be $\frac x2,$ $x,$ or $2x.$ That can be a useful thing to do when you want to evaluate the functions, but we don’t consider this one function the same as the three functions of $x.$

Answer 3

If you introduce an intermediate function $w=bx$ then things may become clearer. We have

$y = \sin(w) = \sin(bx)$

In the $w$ domain the function $y(w)$ repeats with a period of $2\pi$. This period in the $w$ domain is the same regardless of the value of $b$.

But in the $x$ domain the function $y(x)$ repeats with a period of $\frac{2\pi}{b}$. So the period in the $x$ domain does depend on $b$.