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How does $\sin(x-2\pi) = \sin(x)$? Is it so that you can split $\sin(x-2\pi)$ into $\sin(x) - \sin(2\pi)$ and that equals $\sin(x) - 0 = \sin(x)$? Please help. Thank you

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2 Answers

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Sine is a periodic function with period 2$\pi$. Perhaps the easiest way to see this is to use the formula $\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$

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The Sine function has a period of $2\pi$. The period relates to how often the graph goes a full repetition around the unit circle. It is how long it takes sine to return to the same place. Since it starts at $0$, returning to the same place will yield $0$ as well. Thus $\sin 2\pi =0 .$ Expounding on Euler's answer,

$$\sin (x - 2\pi) = \sin x \cos2\pi + \cos x \sin2\pi = \sin x (1) + \cos x (0) = \sin x$$

See more here.

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