# Properties and Graph of the Function y=tan(x)

Properties are following:

1. Domain is x!=pi/2+pik,k in Z (in other words function is not defined for those values of x where cos(x)=0).
2. Range is all number line.
3. Function is periodic with main period pi.
4. Function is odd.
5. Function is increasing on intervals [-pi/2+pik,pi/2+pik],k in Z.
6. Lines x=pi/2+pik,k in Z are vertical asymptotes.

Let's first draw graph on the interval [0,pi/2). Find some values of function:

• if x=0 then y=tan(0)=0;
• if x=pi/4 then y=tan(pi/4)=1;
• if x=pi/3 then y=tan(pi/3)=(sqrt(3))/3.

Draw these points and connect them with smooth line. We've obtained graph of the functon on interval [0,pi/2).

Since y=tan(x) is odd, then draw part of the graph symmetric about origin to the graph on interval [0,pi/2). We've obtained graph of the function on interval (-pi/2,pi/2).

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