
Trigonometry 



Trigonometric
Functions 
Definitions of
trigonometric functions 
Periodicity of
trigonometric functions 





Definitions of
trigonometric functions 
Let x
be an arc of the unit circle measured counterclockwise from
the
xaxis. It is at the same time the circular measure of the
subtended central angle
a as is shown in the below figure.

In accordance with
the definitions of trigonometric functions 
in a rightangled
triangle, 
 the sine of an angle
a
(sina)
in a right triangle is the ratio
of the side opposite the angle to the
hypotenuse. 
 the
cosine of an angle
a
(cosa) in a
right triangle is the ratio of the side adjacent to it to the hypotenuse. 
Thus, from the
right triangle OP′P,
follows 
sinx
= PP′
The sine of arc x
is the ordinate of the arc 
endpoint. 
cosx
= OP′ The cosine of arc
x
is the abscissa of the arc 
endpoint. 




The tangent of an angle
a
(tana)
in a right triangle is the ratio of the lengths of the opposite to the adjacent side. 
The cotangent is defined as reciprocal of the tangent,
thus 

From the similarity of the triangles
OP′P
and OP_{1}S_{1}, 




Hence, the definition of the tangent function in the unit
circle, 
tanx
= P_{1}S_{1}
The tangent of an arc x
is the ordinate of intersection of the second or terminal side (or its 
extension) of the given angle and the tangent line x =
1. 
From the similarity of the triangles,
OP′P
and OP_{2}S_{2}, 


cotx
= P_{2}S_{2}
The cotangent of an arc x
is the abscissa of intersection of the second or terminal side (or its 
extension) of the given angle and the tangent
y
= 1. 
It is obvious from the definitions that the tangent function is not defined for arguments
x
for which cos x =
0,
as well as the cotangent function is not defined for the arguments for which
sin x = 0. 

Periodicity of
trigonometric functions 
After the argument (arc)
x
passes through all real values from the interval 0
< x < 2p or after the terminal side
of an angle turned round the origin for an entire circle, trigonometric or circular functions will repeat their
initial values. 
As the terminal point
P
of an arc continue rotation around a unit circle in the positive direction passing over
the initial point P_{1} , it takes next values from the interval
2p
<
x < 4p, then the values from the interval
4p
< x < 6p and so on. 
On the same way we can examine the rotation of the terminal point
P of an arc
x in the
negative (clockwise) direction, when it will pass through the values from the
intervals, 0 to
2p, from
2p
to 4p, and so on. 
It follows that the argument
x can take any
value, 
x
= a^{rad}
+ k · 2p,
k
= 0, ±1,
±2,
±3,
. . .
or x
= a°
+ k · 360°, k
Î
Z. 
that is, every real value between

oo
and +
oo. 
Particularly, for
k = 0, i.e., during the first rotation the value of
argument is x
= a^{rad}. 
While the arc endpoint continues rounding over the starting point the trigonometric functions will, in
every interval of length
2p (i.e., from
2p to
4p, from
4p to
6p,
. . . , or from
0 to
2p, from
2p to
4p,
. . . ) take the same values in the same order they took in the first interval
[0,
2p]. 
Functions which have the characteristic to take the same values while their argument changes for all integral
multiples of a constant interval (or a constant
increases in amount called increment) we call
periodic functions, and this constant interval we call
period. 
Hence, we say that trigonometric functions are periodic functions of
x, so that 
f (x) = sin x
and
f (x)
= cos x of
the period P
= 2p, 
while
functions, f
(x)
= tan x
and f
(x)
= cot x of
the period P =
p. 
The periodicity of trigonometric functions show the
identities,


sin
(a^{}
+ k · 2p)
= sin a
and cos
(a
+ k · 2p)
= cos a,
k
Î Z 



tan
(a
+ k · p)
= tan a
and cot
(a
+ k · p)
= cot a,
k
Î Z 










Precalculus
contents C 



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