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
  Df  = R \ {(2k + 1) p/2,  k 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 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.
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
  Df  = R \ {kpk 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 + kpk 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 (x1) > f (x2 for all  x1 < x2.
The vertical asymptotes are,   x = kp,   k Z.
Pre-calculus contents G
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