
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 
The
Graph of the Cotangent Function f
(x) = cotx 
Properties of the
cotangent function 
Domain and range 
Zeros of the cotangent
function 
Parity and periodicity
of the cotangent function 
The
cotangent function behavior
and monotony 





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.


The
graph of the cotangent function f
(x) = cot
x 
By associating the values of the cotangent of arcs of the unit circle, to corresponding arcs in a coordinate system obtained are points
P(x,
cot x)
of the graph of the cotangent function.

The graph of the cotangent function in
the down figure is drawn using the relation between tangent and cotangent
which states,

cot
x = 
tan (x + p/2). 


Properties of the
cotangent function 
 Domain 
From the definition,
cot x = cos
x/sin
x, follows that all real numbers
x, as input values, associate exactly
one functional value as output, except the zeroes of the sine function from the denominator, thus 
D_{f} = R \ {kp,
k Î
Z}.

 Range
f (D) = R
or 
oo
< cot x < + oo. 
 Zeros of the
cotangent
function 
The zeroes of the cotangent are determined by the zeroes of the cosine function
from the numerator, thus 
x = p/2
+ kp,
k Î
Z.


Parity
and periodicity 
The cotangent is even function since 

The cotangent is periodic function with the period
p = p
since for every arc x
from the domain 
cot
(x + kp) =
cot x. 

Behavior of the cotangent function
(monotonicity) 
The cotangent is decreasing 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 vertical asymptotes are,
x = kp,
k Î
Z.









Precalculus contents
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