Properties of the Cosine Function

From the graph: cosine is continuous everywhere (no breaks), even (symmetric about the $y$-axis), and $-1\le\cos x\le1$ for all $x$; the maximum $1$ occurs at $x=\ldots,-2\pi,0,2\pi,\ldots$, the minimum $-1$ at $x=\ldots,-\pi,\pi,3\pi,\ldots$

Shift identity. Shifting the graph of $y=\cos x$ right by $\tfrac{\pi}2$ produces the graph of $y=\cos\left(x-\tfrac{\pi}2\right)$, which is identical to $y=\sin x$: $\cos\left(x-\tfrac{\pi}2\right)=\cos\tfrac{\pi}2\cos x+\sin\tfrac{\pi}2\sin x=\sin x$.

Sinusoids in general. For $y=A\sin(\alpha x)$ and $y=B\cos(\beta x)$, always $-A\le A\sin(\alpha x)\le A$ and $-B\le B\cos(\beta x)\le B$, so their amplitudes are $A,B$ and periods $\tfrac{2\pi}{\alpha},\tfrac{2\pi}{\beta}$ respectively — both together called sinusoidal functions. Each is graphed by extending the portion drawn on $\left[0,\tfrac{2\pi}{\alpha}\right]$ (resp. $\left[0,\tfrac{2\pi}{\beta}\right]$). …