# 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.