What does a "mod 2 pi function" mean?
I read somewhere that cosine is a "$\bmod 2 \pi$ function".
I think it means that it repeats every $2\pi$, but what is this "mod"?
$\endgroup$5 Answers
$\begingroup$It is precisely what you said. Given $x,y,a \in \Bbb R$, we can say that $x \equiv y \pmod a$ if $x-y = ka$, with $k \in \Bbb Z$. So saying that $\cos$ (also $\sin$) is a ${\rm mod}\,2\pi$ function means that $\cos(x) = \cos(y)$ if $x \equiv y \pmod {2\pi}$.
$\endgroup$ $\begingroup$"Mod" means for "modulo", the usual arithmetical word. We note, for example, $$11\equiv1[10]$$ to say $11$ equals $1$ modulo $10$. Here, what you are saying is that if $a\equiv b[2\pi]$ then $\cos(a)=\cos(b).$
$\endgroup$ $\begingroup$"mod" is short for "modulo". This was originally used for integers, but it has been generalized to any function that has any kind of period, such as elliptic functions.
As you say, it means the function repeats every $2 \pi$.
$\endgroup$ $\begingroup$Exactly as you said, this refers to the period of cosine and how the values repeat after $2\pi$.
It is also a way to determine if two angles are the same because if their remainders are the same when you divide them by $2\pi$ then they will have the same value when plugged into cosine.
Hopefully this helps :)
$\endgroup$ $\begingroup$Yet another answer and another way to think of it. "mod" stands for "modulo". A "modulo class k" is a set of numbers {a, a + k, a-k, a + 2k, a - 2k} = {a + nk| n $\in$ Z}. To say something is a "mod 2$\pi$ function" (which is pretty casual terminology, IMO) means that all f(a'), where a' are in the same modulo class 2$\pi$, will have the same value.
Which is what everyone else has said. Which is, in this case, synonymous with "has a period 2$\pi$".
Usually we use the term as "a is congruent to b modulo c" which we write as "$a \equiv b \mod c$" which means "a and b are in the same c modulo class".
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