Tangent function is one of the six trigonometric functions. Tangent is abbreviated by tan. Tangent of an angle is described in the context of right triangle like other trigonometric functions. Let us suppose a right triangle as shown in the following figure:
Tangent Graph
Then, tangent of angle $\theta$ is described by the formula:

Tangent Formula
Few selected important values of tan $\theta$ are listed below in the table:
$\theta$  tan $\theta$
0$^o$
30$^o$
$\frac{1}{\sqrt{3}}$
45$^o$
1
60$^o$
$\sqrt{3}$
90$^o$
$\infty$

Following are few important properties of tangent function:
  • Tangent is a discontinuous function.
  • Tangent function is an odd function, since tan (-x) = - tan x.
  • It is a periodic function with period $\pi$.
  • Domain of tangent function is a set of real numbers except for $n\pi + $$\frac{\pi }{2}$, where n is an integer.
  • Range of tangent function is a set of all real numbers, i.e. (-$\infty$, $\infty$).
  • Tangent function has vertical asymptotes - $n\pi + $$\frac{\pi }{2}$, where n is an integer.
  • Reciprocal of tan $\theta$ is cot $\theta$.

Few basic formulas related to tangent function
are as follows:

  1. $\tan \theta$ = $\frac{\sin \theta }{\cos \theta }$
  2. $\tan (A\pm B)$ = $\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$
  3. $\tan 2A$ = $\frac{2\tan A}{1 - \tan^{2}A}$
Many functions are periodic. It means that they repeat themselves after a certain interval. This interval is called the period of the function and such functions are known as periodic functions. Tangent function is a periodic function. Its has a period of $\pi$. The shape of the graph is repeated after every interval of $\pi$.
Graph of tangent has vertical asymptotes at every interval of $\pi$. The graph of a tangent function is given below:
Graph of Tangent
Inverse tangent function is denoted by tan-1 x or arctan x. Domain of inverse tangent is all real numbers, while its range is restricted and is ($-\frac{\pi }{2}$, $\frac{\pi }{2}$). Inverse tangent graph is demonstrated as follows:
Inverse Tangent Graph
Negative tangent graph is shown below:
Negative Tangent Graph
Negative tangent graph is the reflection of positive tangent graph.