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)`.graph of the function y=tan(x)

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.