, 4.1 Angles and Radian Measure
Angles
An formed that have endpoint One called initial side other terminal
angle is
by two
rays a common .
ray is the and the the side .
r c A 7
• ① •
f
y a
Terminal • Initial
side
fB side
vertex
in standard if its ver tex is of system and initial side lies the
An
angle is position at the
origin a
rectangular coordinate its
along
positive x axis
-
r
Tejimiena't
say• y
vertex
↳
•
>
1 a
q
vertex Initial side
positive
Terminal
side
for Initial side
along positive
X -
axis L along
x-axis
there of
when we see an initial side and a terminal side in place , are two kinds rotation that could have
generated the
angle .
Positive angles are
generated by counterclockwise rotation . Thus
, angle at is positive .
Negative angles are
generated by clockwise rotation .
Thus Or
,
angle is
negative .
An is called quadrantal angle if its terminal side lies the axis
Angle B is example of quadrantal angle
angle a on x or
y-axis an a
-
. .
B
Q ✓
Measuring Angles Using Radians
An the of the called The radian of central of
angle whose ver tex is at center circle is a central
angle .
measure
any angle a circle is
circle 's
the
length of the intercepted arc divided by the radius .
Definition of a Radian
One radian is the measure of the central
angle of a circle that intercepts an arc equal in
length to the radius of the circle .
7 fennsthoa
tercepted
← arc
or g.
① =
✓ ← radians
circles
radius
r
• s
r
,Radian Measure
Example :
Computing Radian Measure
A central
angle ,
Q
,
in a circle of radius 12 feet intercepts an arc of length 42 feet .
What is the radian measure of -0 ?
42
E
⑦ =
12 radians
0
= 3.5 radians
•
12
Conversion between Degree and Radians
Example Example
convert each
angle in
degrees to radians : Convert each
angle in radians to degrees .
TX
1841
o
Y ¥ 181 450
-
=
6,10
=
a .
.
or =
radians a .
4 radians •
* red
=
'
I
7o= -4-1381=-2400
'
¥
Eo
÷
=
-
b. 270 .
,
=
2 radians b . 3 radians
•
trad
% -S 1800 6 180
-
-
-30,04 o
=
I 343.80
TE
= =
c .
.
,
=
radians c .
6 radians .
Drawing Angles in Standard Position
figure that full 2x
The illustrates when the terminal side makes one revolution
,
it forms an
angle whose radian measures .
The
figure shows
the
quadrantal angles formed by 3/4
,
1/2
,
and 1/4 of a revolution .
I
¥34 !%!!
- l
2 revolution -4
l revolution revolution
2x radians Sradians y
'
z
2"
a radians ty 2x Iz
⑤
=
-
radians
µ
-
-
-
a
• s s
v
, Example
Draw and label the
angle in standard position : -0 =
-
IT Draw and label the
angle in standard position : Al = IT
'8¥°= If '÷=3"
'
-
E, .
-
=
-
ago 3 .
=
135
r
ya •
•• > >
to
]
13T
Draw and label the
angle in standard position : B =
-7÷ Draw and label the
angle in standard position :
j
= T
3.ae#--sgso
'
If -7 II.
'
-
.
= =
ago
= -
900
4500
7
(
r
B
y 1800
5400
@J
> 00
7200
360°
u
2700 6300
Degree and Radian Measure of Angles Commonly Seen in
Trigonometry
Angles
An formed that have endpoint One called initial side other terminal
angle is
by two
rays a common .
ray is the and the the side .
r c A 7
• ① •
f
y a
Terminal • Initial
side
fB side
vertex
in standard if its ver tex is of system and initial side lies the
An
angle is position at the
origin a
rectangular coordinate its
along
positive x axis
-
r
Tejimiena't
say• y
vertex
↳
•
>
1 a
q
vertex Initial side
positive
Terminal
side
for Initial side
along positive
X -
axis L along
x-axis
there of
when we see an initial side and a terminal side in place , are two kinds rotation that could have
generated the
angle .
Positive angles are
generated by counterclockwise rotation . Thus
, angle at is positive .
Negative angles are
generated by clockwise rotation .
Thus Or
,
angle is
negative .
An is called quadrantal angle if its terminal side lies the axis
Angle B is example of quadrantal angle
angle a on x or
y-axis an a
-
. .
B
Q ✓
Measuring Angles Using Radians
An the of the called The radian of central of
angle whose ver tex is at center circle is a central
angle .
measure
any angle a circle is
circle 's
the
length of the intercepted arc divided by the radius .
Definition of a Radian
One radian is the measure of the central
angle of a circle that intercepts an arc equal in
length to the radius of the circle .
7 fennsthoa
tercepted
← arc
or g.
① =
✓ ← radians
circles
radius
r
• s
r
,Radian Measure
Example :
Computing Radian Measure
A central
angle ,
Q
,
in a circle of radius 12 feet intercepts an arc of length 42 feet .
What is the radian measure of -0 ?
42
E
⑦ =
12 radians
0
= 3.5 radians
•
12
Conversion between Degree and Radians
Example Example
convert each
angle in
degrees to radians : Convert each
angle in radians to degrees .
TX
1841
o
Y ¥ 181 450
-
=
6,10
=
a .
.
or =
radians a .
4 radians •
* red
=
'
I
7o= -4-1381=-2400
'
¥
Eo
÷
=
-
b. 270 .
,
=
2 radians b . 3 radians
•
trad
% -S 1800 6 180
-
-
-30,04 o
=
I 343.80
TE
= =
c .
.
,
=
radians c .
6 radians .
Drawing Angles in Standard Position
figure that full 2x
The illustrates when the terminal side makes one revolution
,
it forms an
angle whose radian measures .
The
figure shows
the
quadrantal angles formed by 3/4
,
1/2
,
and 1/4 of a revolution .
I
¥34 !%!!
- l
2 revolution -4
l revolution revolution
2x radians Sradians y
'
z
2"
a radians ty 2x Iz
⑤
=
-
radians
µ
-
-
-
a
• s s
v
, Example
Draw and label the
angle in standard position : -0 =
-
IT Draw and label the
angle in standard position : Al = IT
'8¥°= If '÷=3"
'
-
E, .
-
=
-
ago 3 .
=
135
r
ya •
•• > >
to
]
13T
Draw and label the
angle in standard position : B =
-7÷ Draw and label the
angle in standard position :
j
= T
3.ae#--sgso
'
If -7 II.
'
-
.
= =
ago
= -
900
4500
7
(
r
B
y 1800
5400
@J
> 00
7200
360°
u
2700 6300
Degree and Radian Measure of Angles Commonly Seen in
Trigonometry
