Properties and Graph of the Function y=sin(x)

Properties are following:

1. Domain is all number line.
2. Range is segment `[-1,1]`.
3. Function is periodic; main period is `2pi`.
4. Function is odd.
5. Function is increasing on intervals `[-pi/2+2pin,pi/2+2pin]` and decreasing on intervals `[pi/2+2pin,(3pi)/2+2pin],n in Z` (see figure).

Let's first draw graph on the interval `[0,pi]`.

Find some values of function:

• if `x=0` then `y=sin(0)=0`;
• if `x=pi/6` then `y=sin(pi/6)=1/2`;
• if `x=pi/2` then `y=sin(pi/2)=1`;
• if `x=pi` then `y=sin(pi)=0`.

Draw these points and connect them with smooth line. We've obtained graph of the functon on interval `[0,pi]`.

Since `y=sin(x)` is odd, then draw part of the graph symmetric about origin to the graph on interval `[0,pi]`. We've obtained graph of the function on interval `[-pi,pi]`.

Now, using the fact that sine is periodic with period `2pi` we can draw graph of the function on all domain.