
Trigonometry 


Trigonometric
Functions 
The unit circle or
the trigonometric circle 
Division of the circumference of the unit circle to
the characteristic angles 
Definitions of
trigonometric functions 





The unit circle or
trigonometric circle 
A circle of radius r =
1, with the center at the origin
O(0, 0) of a coordinate system, we call the
unit or trigonometric
circle, see the figure below. 
The arc of the unit circle that describes a point traveling anticlockwise
(by convention, clockwise is taken to be negative direction) from the initial position
P_{1}(1,
0) on the xaxis, along
the circumference, to the terminal position P
equals the angular measure/distance x
= a^{rad},
in radians.

An angle is in standard position if its initial side lies along the

positive
xaxis. 
If we take the positive direction of the xaxis as the beginning

of a measurement of an angle (i.e.,
a
= 0^{rad},
both sides of

an angle lie on the
xaxis), and the unit point
P_{1}
as the initial

point of measuring the arc, then the terminal side of an angle,

which passes through the terminal point
P
of the arc,

rotating around the origin
(in any direction) describes different 
angles,
and the terminal point P
corresponding arcs, 
x
= a^{rad}
+ k · 2p,
k
= 0,
±1,
±2,
±3,
. . . 
or
x
= a°
+ k · 360°, k
Î
Z. 



It means that every arc x
ends in the same point P
in which ends the corresponding arc
a.

Thus, at any
point P
on the circumference of the unit circle end infinite arcs
x = a
+ k ·
2p, which differ by the
multiplier
2p, and any number
x associates only one point
P. 

Division of the circumference of the unit circle to
the characteristic angles 
There
is a common division of the circumference of the unit circle to
the characteristic angles or the corresponding
arcs which are the multipliers of the angles, 30°
(p/
6) and 45°
(p/
4). 

We can say that a unit circle is at the same time 
numerical
circle. 
The numerical circle shown in the right figure is formed

by winding the positive part of number line, with the unit 
that equals the radius,
around the unit circle in the 
anticlockwise
direction and its
negative part in clockwise 
direction. 

Therefore, the terms angle, arc and number in the

trigonometric
definitions and expressions are mutually

interchangeable.





Example:
In which quadrant lies second or the terminal side
of the angle x
= 1280°. 
Solution: Dividing the given angle by
360°
we calculate the 
number of rotations, or round angles, described by terminal
side 
of the angle x, and the remaining angle
a° position of
which we 
want to find. 

since
x
= a°
+ k · 360°
then
k = 3
and a
= 200°. 
therefore, terminal side of the angle
x lies in the third quadrant. 




Example:
In which quadrant lies the endpoint of the arc
x
= 
47p/3
of a unit circle. 
Solution: Given
arc 


can be expanded to 




Thus, the endpoint of the arc
x
moved around a unit circle in the clockwise (negative) direction 7 times and described additional arc
a
= 
(5/3)p^{rad},
so its endpoint P
lies in the first quadrant. 

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
(sin a)
in a right triangle is the ratio
of the side opposite the angle to the
hypotenuse. 
 the
cosine of an angle
a
(cos
a) in a
right triangle is the ratio of the side adjacent to it to the hypotenuse. 
Thus, from the
right triangle OP′P,
follows 
sin x
= PP′
The sine of arc x
is the ordinate of the arc 
endpoint. 
cos x
= OP′ The cosine of arc
x
is the abscissa of the arc 
endpoint. 




The tangent of an angle
a
(tan a)
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, 
tan
x
= 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}, 


cot
x
= 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. 








Functions
contents B




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