# уроки геометрії

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• 9

1 - 16

• 1

: , 0 180

.

: , 0

180. 0

180

.

: .

: .

. .

1. .

2. ?

3. , , ?

4. : (3;4), (- 2;2),

(-4;-2), (4; -1).

5. , (0;0), (5

4;

5

3), ( )0;

5

3.

. , . [A = 1].

0

.

• 1 .

,

. -

0 180. XOY.

, ,

.

2.

:

1) R=1.

:

R = 1 .

3. .

. = - , -

, .

= 1, -

. sin , cos tg

: sin = 1

= , cos =

1

= , tg =

.

, .

, sin ,

, cos , tg

: sin = 1

= , cos =

1

= ,

tg =

. (0

180

).

4. , ,

(0 180

):

• (0 180

)

, .

(0 180

90

)

.

: tg = 90 ?

[ tg =

tg 90

=

0

1 .]

5. sin a, cos a i tg a 0, 90 180.

0, 90 180 (1;0), (0;1), (-1;0).

,

sin 0 = 0, sin 90 = 1, sin 180 = 0; cos 0 = 1, cos 90 = 0, cos 180 = - 1;

tg00, tg 180 = 0.

: sin 0 = 0, sin 90 = 1, sin 180 = 0; cos 0 = 1,

cos 90 = 0, cos 180 = - 1; tg00, tg 180 = 0? .

.

1. . , ,

2

1.

.

(0;2

1). l || .

1, 2

1

. 1 .

1, 2

1.

: ?

2.. , 10 .

II .

1) , , , D, , , , , OD, OE

• 35, 70, 115, 130, 165.

2) , , , D, . 3) , 35, 70, 115, 130, 165. .

35 70 115 130 165

sin

cos

tg

: sin 35, cos 35, tg 35?

3. . sin , cos tg , : 1) 0 < < 90;

2) 90 < < 180? .

.

0 < < 90 90<

• 2

: , 0 180

.

: , 0

180. sin (180

o ) = sin , cos (180

o ) = - cos ,

tg (180o ) = - tg , .

.

: .

: .

.

1. .

- ?

- , ,

.

- , 0 < < 180?

- , 0 < < 180?

- sin0? cos 0? sin90? cos90? sin180? cos180?

- sin , 0 < < 180?

- cos , 0 < < 180?

- , ',

? ? ? ?

- , cos < 0?

2. 1. ( ).

, '

. , ,

, 5

1.

: 78.

3. 5 ( ).

1) 4 cos 90o + 2 cos 180

o = 4 0 + 2 1 = 2.

2) cos 0o - cos 180

o + sin 90

o = 1 1 + 1 = 1.

.

1 .

8 , -

sin (90 - ) = cos , cos (90 - ) = sin .

• .

= 0, = 90 ?

2. , - , 0 < < 180 , sin (180 - )

= sin ; cos(180- ) = -cos ;

180 - , 0, 90 180,

(x1 1) N (2; 2) .

01 ONNl (ON

= = 1, M0M1 = NON1 = ). 2 = 1 2 = -x1..

(0 180)

, . , sin(180 - ) = si ,

cos (180 - ) = - cos .

.

sin(180 - ) = si , cos (180 - ) =

= - cos , = 0, = 90, = 180 ?

3. .

, 0 180 90,

cos

sin, tg =

cos

sin. ( cos 90 = 0, tg =

= 90).

, tg(180- ) = - tg ( 9 0 )

tg(180 - ) = )180cos(

)180sin(

o

o

=

cos

sin

= - tg.

4. .

, (0 180)

. , ,

(, = 90).

(, ) .

f () = sin , g() = cos , h() = tg ,

,

.

• . , ,

.

, , '. [

180- , ,

' , sin (180- ) > 0 , cos (l80- ) < 0.]

?

sin = 0 ? cos = 0?

sin 120.

.

1. 2 ().

2. 6.

, : 1) 1; 2) 0?

1) cos = 1, = 0. sin 0

= 0.

2) cos = 0, =90. sin 90

= 1.

8.

[sin 135 = sin 45

=

2

2; cos 135

= - cos 45

= -

2

2; tg 135

= tg 45

= 1.]

3. 11. , .

:

1) cos , sin = 5

3 0 90.

sin2 + cos

2 = 1, cos

2 = 1 - sin

2 = 1 -

2

5

3

= 1 -

25

9=

25

16

0 90, cos = 5

4.

2) cos , sin = 3

1 90 180.

sin2 + cos

2 = 1, cos

2 = 1 - sin

2 = 1 -

2

3

1

= 1 -

9

1=

9

8

90 180, cos = - 9

8 = -

3

22 .

3) cos , sin = 4

3.

sin2 + cos

2 = 1, cos

2 = 1 - sin

2 = 1 -

2

4

3

= 1 -

16

3=

16

13

cos = - 16

13 = -

4

13 cos =

4

13.

4. 14. :

• 1) sin 110 cos 140

= sin 90

(- cos 40

) < 0.

2) sin 80 cos 100

cos 148

= sin 80

( - cos 80

) ( - cos 32

) > 0.

3) ) sin 128 cos

2 130

tg 92

o = sin 52

cos

2 50

92cos

92sin= sin 52

cos

2 50

88cos

88sin

<

< 0.

5. 17. :

1) 2 sin 120 + 4 cos 150 - 2 tg l35; 2) cos 120 - 8 sin2 150 + 3 cos 90 cos 162; 3) cos 180 (sin 135 tg60 - cos l35)2.

1) 2 sin 120 + 4 cos 150 - 2 tgl35 = 2 sin 60 - 4 cos 30 + 2 tg 45 = 2 2

3 -

- 4 2

3 + 2 = - 2

2

3 + 2 = - 3 + 2;

2) cos 120 - 8 sin2 150 + 3 cos 90 cos 162 = - cos 60 - 8 sin

2 30 -

- 3 cos 90 cos 18 = - 2

1 - 8

4

1 - 3 0 = -

2

1 - 2 = - 2

2

1.

3) cos 180 (sin 135 tg60 - cos l35)2 = - 1 ((sin 135 tg60)

2

- 2 sin 135 tg60 cos l35 + (cos l35)2) = -1 ((sin 45 tg60)

2 +

+ 2 sin 45 tg60 cos 45 + (cos 45)2 )= -1 ((

2

2 3 )

2 + 2

2

2 3

2

2 +

+ (2

2)

2 )= -

4

6 - 3 -

4

2 = - 2 - 3 .

V.

1. sin (180 - ), cos (180 - )?

2. ' ?

3. , 0 180 90?

4. tg = 90?

5. f () = sin , g () = cos

h () = tg ?

V.

1 ( . 1 14), 9, 12.

3

: .

: ,

.

: .

: ' .

• .

1.

.

2.

(, ) ?

? ? ?

2. . , = 60,

= 3 3 .

.

' ,

, .

.

.

(, , , )

. -

: 3 3 . ^'

, , ,

. ,

' .

• , ?

,

;

;

. .

' .

ABC = 2 , BAC = 60 , = 3 .

.

':

ABC ;

ABC

; , , ,

,

( ).

, '

2 = AC

2 + AB

2-2AC AB cos A ,

.

. .

. -

- (973-1048 pp.). ,

, ,

' :

, '

, .

, .

2.1 ( ).

.

.

ABC. , ,

2 =

2 +

2 - 2 - cosA. :

1) ; 2) ; 3) .

. < 90,

. , , Z < 90.

BD.

• ABD : BD = sinA, AD = cosA. BDC

: 2 = BD

2 + CD

2 = BD

2 =

2 sin

2 A + (AC

- cos A)2 = AB

2-sin

2 A + AC

2 - 2AC-AB-cosA+AB

2-cos

2A =

= AB2 (sin

2 A + cos

2 A) + AC

2 - 2AC -AB cosA =

= AB2 + AC

2 - 2AB -AC cos A.

,

. ,

, ,

: a2 = b

2 + c

2 - 2bc cosa , b

2 = a

2 + c

2-2ac cos,

2 =

2 + b

2 - 2ab cos .

, , ,

.

.

1. . , , .

1) 2 = ( 2 )

2 +3

2 2 3 2 cos 45

= 11 6 = 5.

= 5 .

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