. . The graph of tangent function Graph of the tangent function is useful to find the values of the function over the repeated period of intervals. The tangent function is odd and hence the graph of y = tan is symmetric with respect to the origin.
Since the period of tangent function is π , we need to determine the graph over some interval of length π . Let us consider the interval − and construct the following table to draw the graph of y = tan , x ∈− x ( in radian ) − π − π − π = tan - - - Now, plot the points and connect them with a smooth curve for a partial graph of y = tan , where − . If x is close to π but remains less than π , the sin x will be close to and cos x will be positive and close to . So, as x approaches to π , the ratio sin x is positive and large and thus approaching to ¥ .
Fig. . O æ è ççç ö ø ÷÷÷÷ in π π asymptote asymptote Inverse - - Therefore, the line x = π is a vertical asymptote to the graph. Similarly, if x is approaching to − π , the ratio sin x is negative and large in magnitude and thus, approaching to −∞ .
So, the line x = − π is also a vertical asymptote to the graph. Hence, we get a branch of the graph of y = tan for − < < as shown in the Fig . . The interval − is called the principal domain of = tan Since the tangent function is defined for all real numbers except at ∈ , and is increasing , we have vertical asymptotes ∈ .
As branches of y = tan are symmetric with respect to x ∈ π , , the entire graph of y = tan is shown in Fig. . . Note From the graph, it is seen that y = tan is positive for < < and p < < p ; y = tan is negative for π < < and < <