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

Properties are following:

  1. Domain is `x!=pik,k in Z` (in other words function is not defined for those values of x where `sin(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 `[pik,pi+pik],k in Z`.
  6. Lines `x=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=pi/4` then `y=cot(pi/4)=1`;
  • if `x=pi/3` then `y=cot(pi/3)=sqrt(3)`;
  • if `x=pi/2` then `y=cot(pi/2)=0`.

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=cot(x)

Since `y=cot(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 cotangent is periodic with period `pi` we can draw graph of the function on all domain.