The Graph of Sine Function

Since $\sin(x+2\pi)=\sin x$ for every real $x$, but $\sin(x+p)\ne\sin x$ for any smaller positive $p<2\pi$, the period of $y=\sin x$ is exactly $2\pi$. So the graph over ANY interval of length $2\pi$ — say $[0,2\pi]$ — is repeated identically over every other interval of that length, e.g. $[-2\pi,0],[2\pi,4\pi],[4\pi,6\pi],\ldots$

Table of values on $[0,2\pi]$:

Plotting and smoothly joining these points gives the shape shown in Fig. 4.4: starting at the origin, $y$ rises from $0$ to $1$ as $x$ goes from $0$ to $\tfrac{\pi}2$; then falls from $1$ down through $0$ to $-1$ as $x$ runs from $\tfrac{\pi}2$ to $\pi$ and on to $\tfrac{3\pi}2$; then rises back from $-1$ to $0$ as $x$ goes from $\tfrac{3\pi}2$ to $2\pi$. This one $2\pi$-long portion is called one cycle of sine, and its amplitude is $1$.

Repeating this cycle on either side of $[0,2\pi]$ produces the full graph over all of $\mathbb{R}$, shown in Fig. 4.5. …