# 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.