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math theory in trigonometry for b class in high school with solved probelms

### Transcript of Μιχαήλογλου...

• 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, -

, - .

-

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

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

.

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

______________________________________________________________

180=

1. :

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

, :

. :

+ = = + = +

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. 105, II. 330, III. 22,5, IV. 7200

D)

(.: 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< < ,

(.: 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< < ,

(.: 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