Trigonometry
     Graphs of Trigonometric Functions
     The Graph of the Sine Function  f (x) = sin x
      Properties of the sine function
         Domain and range
         Zeros of the function
         Extremes, maximum and minimum of the sine function
         Parity and periodicity of the sine function
      Behavior of the sine function
Graphs of trigonometric functions
Visual presentation of changes and behavior of each trigonometric function shows us its graph in the coordinate plane xOy.
A graph of a function is formed by points P (x, f (x)), where the abscissas x belong to the domain and the calculated values of the function f (x) as the ordinates, which are the corresponding values from the range.
The graph of the sine function  f (x) = sin x
The unit circle is divided to arbitrary number of equal parts, for example 12 as on the below figure, which then measure 2p/12 = p/6 each.
On the same way starting from the origin divided is its circumference 2p to the same number of equal parts on the x-axis.
  That way, an arc x becomes the abscissa of a point in a coordinate system. By associating the ordinates of the ending points of the arcs x of the unit circle to the corresponding abscissas x obtained are the points        P (x, sin x)  of the curve named sine curve or sinusoid.
Properties of the sine function
 - Domain and range of the sine function
From the graph in the down figure we see that the sine function is defined for all real numbers x that is, the domain of the function Df  = R.
The graph of the sine function is bounded between lines y = -1 and y = 1. Therefore, the function takes all  values from the closed interval [-1, 1], written range ( f ) or  f (D) = [-1, 1].
 - Zeros of the sine function
 The points of intersection of a function with the x-axis we obtain by solving equation f (x) = 0
sin x = 0,  where  x = kpk Z
that is, the x-intercepts are,  x = 0, p, 2p, 3p, . . .  as shows the above diagram.
 - Extremes maximum and minimum of the sine function
The sine function reaches its maximum value 1 at the points whose abscissas are the solutions of the
equation,    
and its minimal value -1 at the points whose abscissas satisfy the equation
 - Parity and periodicity of the sine function
Parity - A function that change sign but not absolute value when the sign of the independent variable is changed is odd function, that is if
f (-x) = - f (x).
Such a function is symmetrical about the origin, as shows the figure above.
Let examine parity of the sine function f (x) = sin x,
f (-x) = sin (-x) = - sin x = - f (x)  therefore, the sine is odd function. 
The sine function is periodic
A function f (x) that repeats its value for all integral multiples of a constant number p added to the independent variable, is called periodic function with period p. That is, if
f (x) = f (x + n p)n = 1, 2, 3, . . .
For the sine function holds the identity
sin (x + k 2p) = sin xk Z 
that means that all positive and negative multiples of 2p are periods of the sine function but the least (principal) period P = 2p.
Behavior of the sine function
We analyze behavior of a function by moving from left to right, i.e., in the direction of the positive x-axis examining the following characteristics,
- intervals where the function is increasing or decreasing,
- maximums and minimums, and 
- roots or zero function values.
Since trigonometric functions are periodic, it is enough to examine its behavior inside of one period. 
Therefore, for the sine function we examine the interval  0 < x < 2p.
The graph of the sine function shows,
           - when the arc x increases from 0 to p/2 the function sin x increases from 0 to 1,
           - when the arc x increases from p/2 to p the function sin x decreases from 1 to 0,
           - when the arc x increases from p to 3p/2 the function sin x decreases from 0 to -1,
           - when the arc x increases from 3p/2 to 2p the function sin x increases from -1 to 0.
Behavior of a function can also be shown in the tabular form,
Trigonometry contents A
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