
Trigonometry 



Trigonometric
Functions 
Definitions of
trigonometric functions 
Periodicity of
trigonometric functions 
Signs of trigonometric
functions 
The table of signs 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 



Signs of trigonometric
functions

Quadrant I
 Values of trigonometric functions, sine, cosine, tangent and cotangent of any arc from
the first quadrant are all positive as positive are the coordinates of the points,
P,
S_{1}
and S_{2}
that define their values.


Quadrant II
 For arcs from the second quadrant points, P
and S_{2}
both have negative abscissas (see the above figure), so the cosine and cotangent are negative. The ordinate of the terminal point
P
is positive so that the sine is positive while the ordinate of the point
S_{1}
is negative, thus the tangent is negative.


Quadrant III
 As the abscissas and the ordinates of the terminal points P
of arcs from the third quadrant (see the above figure) are negative it follows that cosine and sine functions of these arcs are negative. The ordinates
of the points
S_{1} and the abscissas of the points
S_{2} that belong to the arcs from the third quadrant are positive. Thus, the tangent and cotangent of these arcs are positive.


Quadrant IV
 The functions, sine, tangent and cotangent of the arcs from the fourth quadrant are negative as are the coordinates of the points,
P,
S_{1}
and S_{2}, that belong to them. Only the cosine function of arcs from the fourth quadrant is positive as are the abscissas of points
P that belong to them
(see the above figure).


The table of signs of trigonometric
functions



Example:
To which quadrant belongs the endpoint of an arc
a
if
sin a < 0 and
cot a
> 0.

Solution: The
right figure shows that the ordinate of 
the endpoint P
of an arc from the third quadrant is 
negative, so sin
a < 0
while the abscissa of the

point (in which the extension of the terminal side of 
the angle
a
intersects the tangent y =
1) is positive, 
i.e., cot
a
> 0.

That is in accordance with the signs in the above

table.












Trigonometry
contents A 



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