
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
xaxis.

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 
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
D_{f} = 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 function 
The points of intersection of a function with the
xaxis we obtain by solving equation
f (x) = 0, 
sin
x = 0, where x =
kp,
k Î
Z 
that is,
the xintercepts are,
x = 0, ± p, ±
2p, ±
3p,
¼
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 x, k
Î
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
xaxis
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, 









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