what is the practical application of tangent function?

Question

tangent of an angle shows the gradient of a straight line but what this itself shows when it is graphed?

Answer 1

Expanding on what Arturo has told you, I would add that the concept of the tangent to a curve (function's graph) plays a basic role in the Calculus.  At places where the tangent is level (m=0), you may find the local maximum or minimum value of a function. So the introduction to slope (gradient) at early math courses is a very meaningful concept.

Answer 2

Here is an example:

 

Suppose you want to measure the unknown height h of an object located at a known distance b from you.  From where you are standing, measure the angle from the base of the object to the top of the object.  Call it θ.  Then

 

tanθ = h/b

 

h = b tanθ

 

Using the tangent function helped you find the height.  There are vastly many more applications.