
Trigonometry 

Graphs of
Trigonometric Functions 
The Graph of the
Tangent Function f
(x) = tan
x 
Properties of the tangent function

Domain and range 
Zeros of the tangent
function 
Parity and periodicity
of the tangent function 
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 
D_{f} = 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 =
kp,
k 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 (x_{1})
< f (x_{2})
for all x_{1} <
x_{2}.

The equations of the vertical asymptotes are,
x = p/2
+ kp.










Trigonometry
contents A 



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