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Trigonometry |
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Graphs of
Trigonometric Functions |
The
Graph of the Cosine Function f
(x) = cos
x |
Properties of the
cosine function |
Domain and range |
Zeros of the function |
Extremes, maximum and
minimum of the cosine function |
Parity and periodicity
of the function |
Behavior of the cosine
function |
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Graphs of
trigonometric functions |
Visual presentation of changes and behavior of each trigonometric function shows us its graph in the
coordinate plane
xOy.
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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.
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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. |
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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. |
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Properties of the
cosine function |
- Domain,
x Î
R. |
- Range,
-1
<
y < 1. |
- Zeros of the
function,
x
= p/2 +
kp,
k Î
Z. |
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Abscissas
of maximums,
x
= k · 2p,
k
Î
Z
and minimums
x
= (2 k + 1) · p,
k
Î
Z. |
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Parity
and periodicity, the cosine
is even function since f
(-x) =
cos(-x) = cos
x = f (x). |
The identity, cos
(x + k · 2p) = cos
x, k Î
Z shows that the cosine is periodic
function with the period P
= 2p. |
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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. |
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Functions
contents D |
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