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sin 0 = 0 sin pi/2 = 1 sin pi = 0

then why is it's period referred to as 2 pi in most books?

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    $\begingroup$ So $\sin(3\pi/2)=\sin(\pi/2+\pi)=\sin(\pi/2)=1$? $\endgroup$ Commented Dec 21, 2017 at 15:01
  • $\begingroup$ By the way, its period is $2\pi$ since $\sin(x+2\pi)=\sin x$ for all $x$. $\endgroup$ Commented Dec 21, 2017 at 15:02
  • $\begingroup$ The value $0$ happens to be repeated with a period of $\pi$, but other values do not. $\endgroup$
    – Arnaud D.
    Commented Dec 21, 2017 at 15:27

2 Answers 2

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Have you looked at $\sin(1)$ compared to $\sin(1+\pi)$? (Or almost any other number in place of $1$.) Does that look like what a period should mean?

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A period of a function $f:\Bbb R \to \Bbb R$ is $L > 0$ such that $\forall x (f(x) = f(x+L))$, not $\exists x (f(x) = f(x+L))$.

In English, that reads that the function must be unchanged when you shift $x$ by $L$ for every $x$, not just for one $x$.


Note: "the" period of a function is the minimum such $L$.

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