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    06-Mar-2016
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math theory in trigonometry for b class in high school with solved probelms

Transcript of Μιχαήλογλου...

  • www.askisopolis.gr

  • 11

    1.

    1.1. ,

    . - , .

    (0 < < 90)

    : AB

    =

    =

    =

    =

    , 90 D Oxy.

    x (- ) z , .

    2 2(OM ) (x ) (y ) 0 = = + > y x y x , , , x y = = = = .

    Ox - Oz (- ), . (x,y)

    Oz 2 2(OM) x y 0= = + > , :

    y x y x , , (x 0) , (y 0) x y

    = = = = 360. Ox -

    , 360 +D . , 360 +D . -

  • 12

    z , .

    ] :

    ( 360 ) + =D ( 360 ) + =D ( 360 ) + =D ( 360 ) + =D

    , Oxy, .

    1 (0,0) .

    () 1= = y x y , x1 1

    = = = = . y y -

    x x -.

    : 1 1 , 1 1

    1 . -

    y x

    = = ,

    x ZHy

    = = . 1 2 -. - , , 1 2.

    ,

    .

  • 13

    Oxy, -

    , - .

    -

    (rad) , (O, R), R.

    . 0 rad 180= . 3600 2 rad. S rad S = .

    0 rad, 0 0 360 + 2 , + ] . :

    (2 ) + = (2 ) + = (2 ) + = (2 ) + =

    .

    rad

    0 0 0 1 0

    30 6

    12

    32

    3

    3 3

    45 4

    22

    2

    2 1 1

    60 3

    32

    12

    3 33

    90 2

    1 0 0

    180 0 1 0

    270 32

    1 0 0

    360 2 0 1 0

  • 14

    ______________________________________________________________

    rad , :

    180=

    . rad - .

    1. :

    I. 7 rad6

    II. 75 rad.

    I. 180= 7

    6= :

    7 7 6

    180=

    6 o 6 7 180

    110= = o 210=

    II. o 180= = 75 :

    75 5 12180= = 5

    12=

    ______________________________________________________________ 360

    3600, :360. . 360 = +D D D , Z 0 360

  • 15

    rad

    rad

    , :

    . :

    + = = + = +

    f : -

    .

    f : -

    + .

    2. :

    I. 2790 II. 253

    III. 1776

    I. 2790:360 7 270.

    2790 7 360 270= +D D D . : 2790 270 1= = D D 0 02790 270= =

    2790 270 0= =D D 2790 270 0= =D D II. 25:3 25 8 3 1= + .

    25 8 3 1 1 8 83 3 3 3

    + = = + = +

    8 2 :

    25 3 3 3 2

    = = 25 1 3 3 2

    = = 25 3

    3 3= = 25 3

    3 3 3= =

  • 16

    III. 177:6 177 29 6 3= + . 177 3296 6

    = + . 29 , 177

    6 3 9 3

    6 6 2+ = = .

    : 177 3 3 16 6 2

    = + = = 177 3 0

    6 2= =

    177 3 ,6 2

    = 177 3 06 2

    = = .

    ______________________________________________________________

    . - . .. 2 - (, , ) .

    3. 9 14x2 3< < , x x x x 0 > .

    9 14 24 , 42 2 3 3= + = + .

    x 2

    23

    , 2 ,

    :

    ( )

    x 0x 0 x 0

    x x x x 0x 0 x 0x 0 x 0

    +> < > >< > < >

    .

    2

    32

    x

    .

    .

    02

    12

    3 4

    x

    y

    y

    ++

    + +

  • 17

    4. x 2< < , : 2x x 2xx x x 0 + + < .

    : 2x x 2xx x x 0 + + < ( )2x x 2x 1 x 0 + + < 2x(x 1) x 0 + <

    x 2 2

    2x 0 x(x 1) 0 x(x 1) x 0x 0

    < < +

  • 18

    6. x : I. 2 x 3 5x 3+ < II. 2x 4 9 , = + + \ .

    I. x\ : 2 2 x 3 5x 3 x 5x 6 0+ < + <

    x = 2 5 6 0 + < (1) 2 ( 5) 4 1 6 25 24 1= = =

    1,2

    2 5 12 2 3

    = = ==

    (1) 2

  • 19

    7. rad : I. 210 II. 1845 III. 150 IV. 450.

    (A.: I. 76

    , II. 414

    , III. 56

    , IV. 52

    )

    8. :

    I. 7 rad12

    II. 11 rad6

    III. rad8

    IV. 40rad

    (.: I. 105, II. 330, III. 22,5, IV. 7200

    D)

    9. 900 rad4

    . rad .

    (.: 3 x , y8 8

    = = )

    10. :

    I. 1350 II. 27 rad III. 913

    IV. 20064

    (.: I. 1350 270 1,...= = D D , II. 27 0,...= = III. 91 3 ,...

    3 3 2= = , IV. 2006 3 1,...

    4 2= = )

    11. ( )M 1, 3 , 2 , + ] .

    (.: 3(2 )2

    + = , 1(2 ) ,...2

    + = , 3 )

    12. 5 x 32< < , : x x x x > + .

    13. 3 2

    < < , : 1 3 4 0+ + + > . (: 1

  • 20

    15. : 2A 1 x= , 2B x 2 3= , 4 2 x 3= .

    (. 0 A 1 , 6 B 2 , 3 5 )

    16. 7 25

    = + 13 25

    = , ,] , - .

    (: / 2)

    17. x : 2x 4 4 3 , = + \ .

    18. :

    I. 360 150 , 360 210= = +D D D D II. 7 9 2 , 2 , ,4 4

    = = + ] .

    19. 24

    = + 2 , , = + ] . , - .

    (.: 1 2( ) 2 2= + )

  • 21

    :

    1. 2 2 1+ = ( )

    2. , 0 , 0.

    = =

    3. 1 = 4. :

    22

    2

    1 = +

    22

    1 1

    = +

    _____________________________________________________________

    1 : , :

    () 2 2 1+ = ()

    = , 1

    = , .

    2 : , :

    () 2 21

    1 = +

    ()

    = = . , !!!

  • 22

    20. 3x5

    = 3 x 22< < , x rad.

    : 2

    2 2 2 23 9 x x 1 x 1 x 15 25

    + = + = + =

    2 29 16 4 x 1 x x25 25 5

    = = = .

    3 x 22< < , 4

    x

  • 23

    22. x = 4x 3 x 22< < , x rad.

    : 2

    3x 2 x 2x 4x 24x x 41x xx x= 2

    = <

  • 24

    24. 1213

    = 2< < ,

    rad.

    (.: 513

    = , 125

    = , 512

    = )

    25. x = 2 3 x2

    < < , x xKx x

    += + .

    (.: 6 5K25

    = )

    26. 9 52< < 216 9= ,

    1

    += + .

    (.: 53

    = )

    27. 2 23 x x 0 = x 2< < ,

    x rad.

    (.: 1x2

    = , 3x2

    = , x 3= , 3x3

    = )

    28. 6x 8x 10 = x ,2

    , x.

    (.: 4x3

    = )

  • 25

    _____________________________________________________________ : 1 : , -

    . 2 : (-

    ). 3 : -

    . 4 : -

    .

    , - .

    . 1. :

    2 2 x 1 x= , 2 2 x 1 x= 2. x, x, x x, :

    x xx , xx x

    = =

    3. x x 1 = : 1xx

    = 1xx

    = 4. 1 1 ,

    1 1 - ( )( + ) = 2 2 .

    5. :

    2 22 2

    1 11 x , 1 x x x

    = + = +

    6. : 2 2 2 ( ) 2+ = + 3 3 3 ( ) 3( )+ = + +

    , 2 2 x x 1+ = . 1 2 2 x x+ , :

    2 2 21 2xx x x 2xx (x x) = + = .

  • 26

    29. \ :

    . 3 4= +

    2 4+= +

    . 21

    =

    2= + .

    . 4 : 2 2 1+ = 2 2 2 2

    2 2 3 2 ( 3) ( 2)1 1 4 4 ( 4) ( 4) + + + = + = + + + +

    2 2 2 2 2 2( 3) ( 2) ( 4) 6 9 4 4 8 16 + + = + + + + + = + + 2 10 3 0 = ,

    :

    10 112 10 16 7 10 4 72 2 2

    = = = 5 2 7 5 2 7= + =

    . , 1

    1 0

    2