Trigonometry
Graphs of Trigonometric Functions
The Graph of the Tangent Function  f (x) = tan x  The tangent function behavior and monotony
Graphs of trigonometric functions
The graph of the tangent function  f (x) = tan x
By associating the ordinates of the points that according to definition equals the tangent of an arc in the unit circle, to corresponding arc x in a coordinate system obtained are points P (x, tan x) of the graph of the tangent function. Properties of the tangent function
- Domain and range
From the definition of the tangent, tan x = sin x/cos x, follows that all real numbers belong to the domain of the tangent function except the zeroes of the cosine function, thus
Df  = R \ {(2k + 1) · p/2,  k Î Z}.
As shows the graph in the above figure, tangent function takes all real values from  - oo to + oo  as its argument x passes through an interval of the length p, therefore the range
f (D) = R  or  - oo < tan x < + oo.
- Zeros of the tangent function
The zeroes of the tangent are determined by the zeroes of the sine function in the numerator, so
x = kpk Î Z.
- Parity and periodicity
The tangent is odd function since
f (-x) = tan (-x) = - tan x = - f (x).
It is obvious from the graph that the tangent is periodic function with the period  p = p. Thus, for every arc x from the domain
tan (x + kp) = tan x.
Behavior of the tangent function  (monotonicity - a function consistently increasing or decreasing in value)
The tangent is increasing function in every interval between any of the two successive vertical asymptotes
that is,    f (x1) < f (x2 for all  x1 < x2.
The equations of the vertical asymptotes are,   x = p/2 + kp.   Trigonometry contents A 