chapter-8-circular-measure.pdf

��

��

��

��

1. �� , �� . �� , �� . � � �

1 ��

�

� � 2. �� 1 ��, �� . 3. �� , �� :

1 rad

r

=

2π r

360 1 rad =

r 2π r

× 360

2π rad = 360 ° 4. π rad = 180°

1 rad = =

180°

π 57.3° �� 57°18

'

5. 2π rad = 360°

1° =

π 180

radian

��

�� 120° ��

1° =

π 180

radian

�� 94

��

��

120° = 120 × 2π

=

3

π 180

radian

radian �� 2.0947 ��

�� '

�� 112°36 ��

��

1°

60'

�� '

112°36 = 112° + 36 = 112° + (

'

36

)° 60 = 112° + 0.6° = 112.6°

1° =

π 180

radian

112.6° = 112.6 ×

π 180

radian

= 1.965 ��

π

��

6

rad ��

��

1 rad =

180°

π π π 180° rad = × 6 6 π = 30 °

��

�� 1.36rad ��

1 rad =

180°

π

�� 95

��

��

1.36rad = 1.36 × =

244

180°

π

°

π =

77.92°

��

1. �� . (π = 3.142 ) (�) 0.37 �� (�) 2.04 �� (�) 1.19 �� 2. �� , �� 4 �� .

(π = 3.142) (�) 248°9

'

(�) 304°22 (�) 46°14

'

'

��

�

�

θ �

S

�

θ

=

Circumference of a circle

angle of a whole turn

�� , �� 360° .

S

=

θ

2π r

360 °

S =

θ × 2π r 2π

θ �� .

�� 2π rad = 360 ° �� , �� 2π ��. �� 96

��

��

��,

S

=

Circumference of a circle

θ 2π r 2π 2π r θ S = 2π S = r θ S

θ 2π

=

θ �� .

��

�

7.68��

θ O

�

�� 6 ��. �� 7.68 ��. �� θ , �� . ��

��

S = r θ �� = 6 �� = 7.68 ��,

7.68 = 6θ

θ = 1.28 ��

�� 5 ��. �� ∠POR �� 1.2 ��. ��

��

S = r θ �� = 5 �� θ = 1.2 ��,

S = (5)(1.2) S = 6 �� 97

��

��

��

1. �� 3, �� 32 ��. �

θ �

�

�� 3 (�) �� θ . (�) ��

θ = 1 .2 ��.

2. �� 4.2 �� 1.4 ��. �� 3. �� 2π ��. ��

240° . �� . 4. �� 4 �� . �� = ��= 3�� .

� �

1 3

��

�

�

�

�� 4 �� 12 ��, �� .

��

�

θ �

A Area of circle

=

�

θ angle of a whole turn

�� 98

��

��

�� , �� 360° .

�

θ

=

π � 2

360 °

θ �� .

θ 2 × π � 2π

� =

�� 2π rad = 360 ° �� , �� 2π ��. ��,

θ 2π π r 2 π r 2θ A = 2π A

A

=

1

=

2

θ �� .

2

r θ

�� ,

1

A = =

1 2

A

2

r 2θ

S = r θ

r .r .θ =

1 2

rS

�� . ��

��

1 �

2

�

× base ×

height ��

�� .

�

��

�

�� , ��

�

��. �� =

1 2

× AC ×

h

�

�� 99

��

��

�� =

h

� ��

= ��

��

�

��

��

�

,

�

�� . ��

Area

=

Area

=

Area

=

1 2 1 2 1 2

�� . �� , ��

× AC × AB × sin A

� ��

�� .

× AB × BC × sin B × AC × BC × sin

C

��

�� .

��

� = � =

1 2 1 2

2

� θ −

1 2

2

� sin θ

��

��

2

� (θ − sin θ )

1 2

× �� × �� × sin � ,

��

1 2

��

2

� sin θ ��

��. �

7.68��

θ

�

�

�� 6 ��. �� 7.68 ��. �� θ , �� . ��, �� . ��

��

S = r θ �� = 6 �� = 7.68 ��,

7.68 = 6θ

θ = 1.28 ��

�� 100

��

��

�� =

1

2

( 6 )(1 .28 ) −

2 = 23 .04 =

5 .8��

1 2

2

( 6 )(sin 1 .28 )

− 17 .24

��

2

�� .

��

�� 16 ��. �� : �� = 5: 8. �� ∠ORQ = 90° � �

x �

�

16 cm �� (�) �� , �� . (�) �� , �� cm

2

��

��

OP

=

5 8

OR

�� OR = 16cm .

�� OR = 16 cm ��

OP

=

16

OP =

OP OR

=

��

5 8

.

�� , ��

5

��

8 5

��.

× 16 8 = 10cm

�) �� ∠ORQ = 90°

cos x =

x

10 16

=

−1 10 cos ( ) 16

=

0.8957 rad

�� 101

��

��

�) �� =

1

2

(16 )( 0.8957 ) −

1

2 2 = 114 .6496 − 62 .4517 =

52 . 1979 cm

(10 )( 6 )(sin 0 .8957 ) �� .

2

�� 2

1. �� 4, �� 21 . 6cm . �

θ �

�

�� 4 (�) �� θ . (�) ��

θ = 1 .2 ��.

2. �� 3.6 �� 1.15 ��. �� . 2

3. �� 12π �� . �� 270° . �� . ��

C

Q ( 8, 6 ) B

θ A P ( 8, 4 )

D

E (12 ,4)

�� . ��: (�) �� (�) ��

θ ��

(�) �� (�) ��

�� 102

��

��

�� 2

2

(�) PE = (12 − 8) + (4 − 4)

��

2

PE = 16 ��

PE = ±4 radius > 0 �� = 4 �� (�) cos ∠PBX = ∠PBX = cos

−1

1

� ��

4

��. �� θ �� .

 1     4 

�� . �� θ .

= 1.1181rad

θ = 1.3181× 2 =

2.6362rad

(�) �� = =

1 2

( 4 2 )( 2 .6362 )

21 .0896 unit

2

1  ( 4 )( 4)(sin 2. 6362 )  2 

(�) �� = 21 .0896 − 2  =

21 .0986

−

7 .7464

= 13 . 3442 cm

2

�� . �� . �� .

��

1. �� . �� 4 ��.

P

O

Q �� 20 ��, �� .

�� 103

��

��

2

2. �� 19 �� 22.5 �� . �� . 2

3. � �� 4.5 ��. �� 135 ° , �� . (π = 3 .142 ) 4. �� 40 ° . �� . (π = 3 .142 ) 5. �� , � �� . �� 30 ��.

S R

Q

8cm

β

5 cm (� ) �� α �� β

P

O 2

(�) �� α �� β �� , �� 10 ��

6. �� 40 ��.

P

�� (�) �� (�) ��

42° O

r

Q

�� 104

chapter-8-circular-measure.pdf

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