Trigonometry
Graphs of Trigonometric Functions
The Graph of the Cosine Function  f (x) = cos x
Properties of the cosine function
Domain and range

Parity and periodicity of the cosine 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 cosine function  f (x) = cos x
To draw the graph of the cosine function divide the unit circle and x-axis of a Cartesian coordinate system the same way as when drawing the sine function.
The abscissas of the ending points of arcs x, of the unit circle, are now the ordinates of the corresponding points P(x, cos x) of the graph, as shown in the figure below.
Since   cos x = sin (x + p/2) that is, the cosine of an arc x equals the sine of the same arc increased by       p/ 2.
Therefore, the graph of the cosine function correspond to the graph of the sine function translated in the negative direction of the x-axis by p/ 2.
Thus for example,  cos p/6 = sin (p/6 + p/ 2) = sin 2p/3 as shows the above graph.
Properties of the cosine function
- Domain,    x Î R.
- Range,    -1 < y < 1.
- Zeros of the function,    x = p/2 + kpk Î Z.
- Abscissas of maximums,    x = k · 2pk Î Z  and  minimums    x = (2 k + 1) · pk Î Z.
- Parity and periodicity, the cosine is even function since  f (-x) = cos(-x) = cos x = f (x).
The identity  cos (x + k · 2p) = cos xk Î Z  shows that the cosine is periodic function with the period  P = 2p.

Behavior of the cosine function
The table shows behavior of the cosine function while the arc x of the unit circle increases passing through all the values from the period.
Pre-calculus contents G