# Period of the sum/product of two functions

Suppose that period of $f(x)=T$ and period of $g(x)=S$, I am interested what is a period of $f(x) g(x)$? period of $f(x)+g(x)$? What I have tried is to search in internet, and found following link for this.

Also I know that period of $\sin(x)$ is $2\pi$, but what about $\sin^2(x)$? Does it have period again $\pi n$, or? example is following function $y=\frac{\sin^2(x)}{\cos(x)}$ i can do following thing, namely we know that $\sin(x)/\cos(x)=\tan(x)$ and period of tangent function is $\pi$, so I can represent $y=\sin^2(x)/\cos(x)$ as $y=\tan(x)\times\sin(x)$,but how can calculate period of this?

• The period of $\sin(x)$ is not $\pi n$, but rather $2 \pi$. See this plot on W|A. Jun 28, 2012 at 15:20
• right thanks,thanks for correction Jun 28, 2012 at 15:22
• Just use the definition of the period. Jun 28, 2012 at 15:25
• try $lcm$ of periods.
– Aang
Jun 28, 2012 at 15:28
• @dato: There is never any case where gcd is relevant here (except perhaps sometimes by accident). Jun 28, 2012 at 15:42

We make a few comments only.

1. Note that $$2\pi$$ is a period of $$\sin x$$, or, equivalently, $$1$$ is a period of $$\sin(2\pi x)$$.
But $$\sin x$$ has many other periods, such as $$4\pi$$, $$6\pi$$, and so on. However, $$\sin x$$ has no (positive) period shorter than $$2\pi$$.
2. If $$p$$ is a period of $$f(x)$$, and $$H$$ is any function, then $$p$$ is a period of $$H(f(x))$$. So in particular, $$2\pi$$ is a period of $$\sin^2 x$$. However, $$\sin^2 x$$ has a period which is smaller than $$2\pi$$, namely $$\pi$$. Note that $$\sin(x+\pi)=-\sin x$$, so $$\sin^2(x+\pi)=\sin^2 x$$. It turns out that $$\pi$$ is the shortest period of $$\sin^2 x$$.
3. For sums and products, the general situation is complicated. Let $$p$$ be a period of $$f(x)$$ and let $$q$$ be a period of $$g(x)$$. Suppose that there are positive integers $$a$$ and $$b$$ such that $$ap=bq=r$$. Then $$r$$ is a period of $$f(x)+g(x)$$, and also of $$f(x)g(x)$$.

So for example, if $$f(x)$$ has $$5\pi$$ as a period, and $$g(x)$$ has $$7\pi$$ as a period, then $$f(x)+g(x)$$ and $$f(x)g(x)$$ each have $$35\pi$$ as a period. However, even if $$5\pi$$ is the shortest period of $$f(x)$$ and $$7\pi$$ is the shortest period of $$g(x)$$, the number $$35\pi$$ need not be the shortest period of $$f(x)+g(x)$$ or $$f(x)g(x)$$.
We already had an example of this phenomenon: the shortest period of $$\sin x$$ is $$2\pi$$, while the shortest period of $$(\sin x)(\sin x)$$ is $$\pi$$. Here is a more dramatic example. Let $$f(x)=\sin x$$, and $$g(x)=-\sin x$$. Each function has smallest period $$2\pi$$. But their sum is the $$0$$-function, which has every positive number $$p$$ as a period!
4. If $$p$$ and $$q$$ are periods of $$f(x)$$ and $$g(x)$$ respectively, then any common multiple of $$p$$ and $$q$$ is a period of $$H(f(x), g(x))$$ for any function $$H(u,v)$$, in particular when $$H$$ is addition and when $$H$$ is multiplication. So the least common multiple of $$p$$ and $$q$$, if it exists, is a period of $$H(f(x),g(x))$$. However, it need not be the smallest period.
5. Periods can exhibit quite strange behaviour. For example, let $$f(x)=1$$ when $$x$$ is rational, and let $$f(x)=0$$ when $$x$$ is irrational. Then every positive rational $$r$$ is a period of $$f(x)$$. In particular, $$f(x)$$ is periodic but has no shortest period.
6. Quite often, the sum of two periodic functions is not periodic. For example, let $$f(x)=\sin x+\cos 2\pi x$$. The first term has period $$2\pi$$, the second has period $$1$$. The sum is not a period. The problem is that $$1$$ and $$2\pi$$ are incommensurable. There do not exist positive integers $$a$$ and $$b$$ such that $$(a)(1)=(b)(2\pi)$$.
• Another potential problem can occur when we do have commensurable periods. Consider $f(x)=\sin(x)$ and $g(x)=x-\sin(x)$, for instance. Jun 28, 2012 at 17:04
• @Dato: Division is the same, it is covered by item $4$, where $H(u,v)=u/v$. Jun 29, 2012 at 6:48
• @Dato: It can be figured out. For if $f(x+p)=f(x)$ and $g(x+p)=g(x)$ then $H(f(x+p), g(x+p))=H(f(x),g(x))$. Jun 29, 2012 at 6:57
• Very well written and truly illuminating answer :) Aug 8, 2016 at 13:43
• I would add that even if your two functions have minimal periods which are not commensurable, it is still possible that their sum might be periodic by accident (with a totally unrelated period). Indeed, it takes a decent amount of cleverness to prove that $f(x)=\sin x+\cos 2\pi x$ really isn't periodic. Dec 14, 2016 at 3:12

If you are suppose to find period of sum of two function such that, $$f(x)+g(x)$$ given that period of $$f$$ is $$a$$ and period of $$g$$ is $$b$$ then period of total $$f(x)+g(x)$$ will be $$\operatorname{LCM} (a,b)$$. But this technique has some constrain as it will not give correct answers in some cases. One of those case is, if you take $$f(x)=|\sin x|$$ and $$g(x)=|\cos x|$$, then period of $$f(x)+g(x)$$ should be $$\pi$$ as per the above rule but, period of $$f(x)+g(x)$$ is not $$\pi$$ but $$\pi/2$$. So in general it is very difficult to identify correct answers for the questions regarding period. Most of the cases graph will help.

• But technically, in your example, $\pi$ is still a period of f(x)+g(x), it just so happens not to be the smallest period. Dec 18, 2019 at 14:47