Analysis of the tangent graph for period, range, domain and inverse

Tangent is one of the basic function of trigonometry along with sine and cosine functions. Graph of tangent function is very useful to learn about it.

Analyze the graph of tangent function

Before starting graphs of tangent functions, let us have a small introduction on tangent function. For right-angle triangle, tangent is found by taking ratio of length of opposite side to the length of adjacent side. Mathematically tangent is a ratio of sine function and cosine function.

tanA = sinA / cosA

Tangent graph is very useful for understanding the functionality of tangent. We will soon observe some of properties of tangent function like domain, range and period of a tangent graph by plotting the tangent function graph.

Graph of tangent function is prepared by plotting values of angles (say x) verses value of tangent function for that angle (say tanx). For this first you have to prepare tangent function table like,
tangent function table

Then we can plot graph as shown below,
graph of tangent function

  1. From above graph we can note some properties of tangent function.
  2. tangent function graph repeats after every π periods, so tangent graph period is π.
  3. The range of tangent function graph is (−∞,+∞).
  4. tangent function graph is symmetric around origin (0,0)
  5. tangent function is odd function, so tan(-x) = -tanx.
  6. tangent function's domain is all real values except (π/2+kπ), where k= any integer number.

After finishing the tangent function graph, now we will study the graph of inverse tangent.

Analyze the inverse tangent graph

All trigonometry functions are having their inverse form. So the tangent function also has its inverse form named tan inverse or arctan. As we plot tangent graph, we can also plot inverse tangent graph following same procedure. Find the value of y=tan-1x for some values of x and after plotting it we get the graph of tan inverse as shown below,
inverse tangent graph

Tangent graph transformations can be done in many ways like you can changing the period, shift the graph vertically or horizontally or transform it vertically.

If we have y = atan(bx), then period of tangent is π/b.

For this if a is positive then period of graph increase and if a is negative then period of graph decrease. Let us see one example on this.
Tangent graph transformations

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