
Trigonometry 



Trigonometric
Functions 
Unit of measurement of
angles  a radian (the circular measure) 
Protractor  an instrument for measuring angles 
Degrees to radians and radians to degrees conversion examples 
The unit circle or
the trigonometric circle 
Division of the circumference of the unit circle to
the characteristic angles 





Unit of measurement of
angles  a radian (the circular measure) 
The relation between
a central angle a
(the angle between two radii) and the corresponding arc l
in the circle of radius r is shown by the
proportion, 
a°
: 360° _{}
= l
: 2rp 
It shows that the central angle
a°
compared to the round angle of 360° 
(called
perigon) is in the same relation as the corresponding arc
l 
compared
to the circumference 2rp.
Therefore, 




where the ratio 

we call the
circular measure, usually denoted
a^{rad},
i.e., 


thus, 


The central angle subtended by the arc equal in length to the radius, i.e.
l = r, 

we call it
radian. 
Thus, the angle
a = 1° equals in
radians, 




or
arc1° =
0.01745329. Arc is abbreviation from Latin
arcus,
(p =
3.1415926535...). 

Protractor
 an instrument for measuring angles 
Mentioned relations between units of
measurement of an angle and arc clearly shows the protractor
shown in the
below figure marked with radial lines indicating degrees,
radians and
rarely used gradians (the angle of an entire
circle or round angle is 400 gradians). 

A right angle
equals 100 grad (gradians). 
The hundredth part of a right angle is
1^{g}
grad,
and one 100th part of 1grad is centesimal arc minute 1^{c},
and one 100th part of centesimal arc
minute is centesimal arc second 1^{cc},
therefore 


Degrees
to radians and radians to degrees conversion examples 
Example:
Convert
67° 18´
45" to radians.

Solution: The given angle we write in the expanded notation and calculate its decimal
equivalent, 

then use the formula to convert degrees to radians 

Using a scientific calculator, the given conversion can be performed almost direct. 
Before a calculation
choose right angular measurement (DEG, RAD, GRAD) by pressing
DRG key, then
input, 67.1845 INV ®DEG
67.3125° 
Because a calculator must use
degrees divided into its decimal part one should press ®DEG
(or ®DD) to get
decimal degrees. Then press
INV DRG® to get
radians, 1.174824753^{rad}. 

Example:
Convert 2.785^{rad}
to degrees, minutes and seconds. 
Solution: Using formula, 

The same result one obtains with a calculator through the
procedure, press DRG key to set RAD measurement,
then input 2.785
INV DRG® 177.2986066
grad = 177^{g}29^{c}86^{cc}, press
again INV DRG® 159.5687459° obtained are
decimal degrees (DEG), and to convert to degrees/minutes/seconds press INV
®DMS to
get 159° 34´
7.48". 

Example:
Find the length of the
arc l
that subtends the central angle a
= 123°
38´ 27"
in the circle of
radius r =
15 cm. 
Solution: First express the angle
a
in decimal degrees, i.e. 


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. 









Trigonometry
contents A 



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