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 + (AC AD)

    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 .