
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 






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. 









Functions
contents B




Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 
