Hand graphing trigonometric equations

The current section of my book is an inexcusably wordy dissertation on graphing sine and cosine functions that makes no sense.

The equations given for graphing sin and cos functions are $\displaystyle y=A*sin(b*x)$ and $\displaystyle y=a*sin(b*x)$

A is amplitude and the range of the function is $\displaystyle -|a|,|a|$ (pretend there are square brackets around that)

My issue comes witht he period of the function. When b=1 the period is $\displaystyle 2\pi$, however with how they're teaching it it looks like that when b=2 the equation becomes $\displaystyle 2\pi/2$

So if I've got this right, a more appropriate function is $\displaystyle y=A*sin(2\pi/b*x)$ and the same with the cosine function?

The section goes on to teach to take $\displaystyle 2\pi/b$ and set it at the end of an interval, and divide that interval into four equal parts.

After that it spends another few pages making absolutely no sense, what I know I got from the video lecture which doesn't cover all that the book wants me to know.

So, how do I sketch graphs of sin and cos functions? Anyone have a link to a page that says so in concise terms?