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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?

Please help me.

Cameron Buie
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dato datuashvili
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2 Answers2

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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 $(a)(1)=(b)(2\pi)$.
Marcus Müller
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André Nicolas
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  • Another potential problem can occur when we do have commensurable periods. Consider $f(x)=\sin(x)$ and $g(x)=x-\sin(x)$, for instance. – Cameron Buie Jun 28 '12 at 17:04
  • @CameronBuie: I do not understand the $x-\sin x$, since it does not have a period. – André Nicolas Jun 28 '12 at 17:18
  • /facepalm/ That was supposed to be $1-\sin(x)$. Of course the constant function $f+g$ is then periodic, but it is periodic of all periods. – Cameron Buie Jun 28 '12 at 17:31
  • @CameronBuie: I had already given a very similar example in the answer, with $\sin x$ and $-\sin x$. – André Nicolas Jun 28 '12 at 17:32
  • Ah. I see now that you have. Apologies for wasting your time. – Cameron Buie Jun 28 '12 at 17:34
  • @CameronBuie: Actually, my thanks should be to you. I looked back to see whether my memory was right, and found a bad typo. – André Nicolas Jun 28 '12 at 17:42
  • @André Nicolas thanks very much,what about if we have fraction of two function?can we consider this fraction as inverse of multiplication and for calculating period use rule of multiuplication? – dato datuashvili Jun 29 '12 at 06:45
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    @Dato: Division is the same, it is covered by item $4$, where $H(u,v)=u/v$. – André Nicolas Jun 29 '12 at 06:48
  • thanks for so fast comment,i understood everything,unfortunately in my book where this question was asked,there was not explained such situation such as period of two function multiplication,division and so on – dato datuashvili Jun 29 '12 at 06:51
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    @Dato: It can be figured out. For if $f(x+p)=f(x)$ and $g(x+p)=g(x)$ then $H(f(x+p), g(x+p))=H(f(x),g(x))$. – André Nicolas Jun 29 '12 at 06:57
  • @dato: You are welcome. Things are not too complicated as long as we are talking about a period. But we can construct weird functions $f$ that have multiple periods that cannot be compared easily. – André Nicolas Jun 29 '12 at 15:43
  • @curryage: I am typo-prone, but am relieved there isn't a typo in the second property. I really mean $H(f(x))$. In the specific example, we have $f(x)=\sin(x)$ and $H(y)=y^2$. – André Nicolas Apr 24 '14 at 05:52
  • @AndréNicolas: In regard to point 4, is there any confirmation that the sum of 'simple' sinusoids, of the form $Asin(mx) + Bsin(nx) + ...$ will have it's 'smallest' period equal to the LCM of periods of each constituent sinusoid? – Sidd Jul 04 '15 at 15:27
  • Yes, for $A$ and $B$ both non-zero, and integers $m$ and $n$, the (smallest) period can be computed using the lcm. – André Nicolas Jul 05 '15 at 01:32
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    Very well written and truly illuminating answer :) – Max Payne Aug 08 '16 at 13:43
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    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. – Eric Wofsey Dec 14 '16 at 03:12
  • what about fractional periods? so if f(x) has period a/b and g(x) has period c/d, where a,b,c,d are integers what are the periods of (i) f(x)+g(x), (ii) f(x)*g(x) ? – spiros Jan 21 '24 at 23:53
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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.

Ted Hopp
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Chirag Palan
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