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Radian and degree measures of angles
A degree measure.
Here a unit of measurement is a degree (its designation is ° or deg ) – a turn of a ray by the 1 / 360 part of the one complete revolution. So, the complete revolution of a ray is equal to 360 deg. One degree is divided into 60 minutes (a designation is ‘ or min); one minute – correspondingly into 60 seconds (a designation is “ or sec).
A radian measure.
As we know from plane geometry ( see the point “A length of arc” of the paragraph “Geometric locus. Circle and circumference” ), a length of an arc
l
, a radius
r
and a corresponding central angle are tied by the relation:
= l /
r
.
This formula is a base for definition of a radian measure of angles. So, if
l
=
r
, then
= 1, and we say, that an angle
is equal to 1 radian, that is designed as
= 1 rad. Thus, we have the following definition of a radian measure unit:
A radian is a central angle, for which lengths of its arc and radius are equal
( A
m
B = AO, Fig.1 )
. So, a radian measure of any angle is a ratio of a length of an arc drawn by an arbitrary radius and concluded between sides of this angle to the arc radius.
Following this formula, a length of a circumference
C
and its radius
r
can be expressed as:
2
=
C / r
.
So, a round angle, equal to 360° in a degree measure, is simultaneously 2
in a radian measure. Hence, we receive a value of one radian:
Inversely,
It is useful to remember the following comparative table of degree and radian measure for some angles, we often deal with:
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