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| Trigonometry |
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Graphs of
Trigonometric Functions |
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The Graph of the
Tangent Function f
(x) = tan
x |
Properties of the tangent function
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Domain and range |
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Zeros of the tangent
function |
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Parity and periodicity
of the tangent function |
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The
tangent function behavior
and monotony |
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The
Graph of the Cotangent Function f
(x) = cotx |
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Properties of the
cotangent function |
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Domain and range |
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Zeros of the cotangent
function |
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Parity and periodicity
of the cotangent function |
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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 |
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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 |
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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 |
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f(D) = R
or -
oo < tan x < + oo. |
| - Zeros of the tangent
function |
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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). |
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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 |
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By associating the values of the cotangent of arcs of the unit circle, to corresponding arcs in a coordinate
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| 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
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| 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 B |
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