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Trigonometry |
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Graphs of
Trigonometric Functions |
The Graph of the
Tangent Function f
(x) = tan
x |
Properties of the tangent function
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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 |
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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. |
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Properties of the tangent function
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- 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}.
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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.
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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. |
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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.
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The equations of the vertical asymptotes are,
x = p/2
+ kp.
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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.
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The graph of the cotangent function in
the down figure is drawn using the relation between tangent and cotangent
which states,
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cot
x = -
tan (x + p/2). |
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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 |
Df = R \ {kp,
k Î
Z}.
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- 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.
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Parity
and periodicity |
The cotangent is even function since |
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The cotangent is periodic function with the period
p = p
since for every arc x
from the domain |
cot
(x + kp) =
cot x. |
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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 (x1)
> f (x2)
for all x1 <
x2.
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The vertical asymptotes are,
x = kp,
k Î
Z.
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Pre-calculus contents
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