תורעה – תוירטמונוגירט תויצקנופ – 7 ליגרת

Mathematics, Summer 2011 / Exercise 7 ? Notes

? ? 7

! ? . .1

2 sin

2

-

x

= 1 ()

tan x = 1 ()

cos x

=

1 2

()

sin x

=

1 2

()

cos x

=

sin

x 2

()

4 sin x cos x =

3 ()

sin (2x) + sin x = 0 ()

sin x = tan x ()

cos (2x) = 2 cos x sin x () tan2 x - 1 = 0 ()

cos x =

1 tan x

()

sin2 x - 1 = 0 ()

cos ( + x) = sin ( - x) ()

sin x

=

cos

x 2

()

sin2 x = 2 sin x - 1 ()

cos2 x - 1 = 0 ()

, -

6

,

,

.x =

6

()

: .2k

5

+ 2k or

+ 2k

6

6

,-

3

,

,

.x

=

3

()

: .2k

+ 2k

or

- + 2k

3

3

: ()

tan x = 1 sin x

=1 cos x sin x = cos x

.

5 4

4

:

?sin x = cos x ,

.

4

+

k

,

. , tan x

,k

(

4

)

tan x

=

1

.

4

+ k

:

1

Mathematics, Summer 2011 / Exercise 7 ? Notes

sin - x = cos x 2 1 cos x = 2

: ()

. : ()

sin x = tan x sin x

sin x = cos x

: ?sin x = 0 ,sin x

k

,

0=

0 cos 0

.x = k

.sin x = 0

.

.sin x = 0

1 1=

cos x cos x = 1

. . sin x = 0 x = 2k .k : ()

2 sin x cos x + sin x = 0 sin x (2 cos x + 1) = 0

. k ,x = k sin x = 0 .2 cos x + 1 = 0 sin x = 0 ,

.cos x

=

-

1 2

2 cos x + 1

=

0

2 cos x = + 2k

or

cos x = - 2 + 2k

3

3

,

.-

2 3

+ 2k

2 3

+ 2k

k

.

. ()

2

Mathematics, Summer 2011 / Exercise 7 ? Notes

. ()

:

.(x

,

x 2

)

,

cos2 x - sin2 x

=

x sin

2

2

2

,y

sin

x 2

,

. .

.sin x = -1 sin x = 1 , .(! ) sin x = ?1 ,sin2 x = 1 ()

. ,

: () 1 cos x = tan x cos x cos x = sin x : ? ,cos x ,

: cos x

cos x 1 - 1 = 0 sin x

.(

,) x =

2

+

k

.1

-

1 sin x

= 0 cos x = 0

1 1=

sin x sin x = 1

.x

=

2

+ k

.x

=

2

+ 2k

.tan x = -1 tan x = 1 , .(! ) tan x = ?1 tan2 x = 1 ()

. , .

cos x = , 2x .cos (2x) = sin (2x) () ( ) sin x

2x = + k

4

x= + k 82

. () 3

Mathematics, Summer 2011 / Exercise 7 ? Notes

: ()

sin2 x - 2 sin x + 1 = 0

: ! ? .a2 - 2a + 1 , (sin x - 1)2 = 0

.x

=

2

+ 2k

,sin x

=

1

,sin x - 1

=

0

:

x 2

+

x 2

x

()

xx

x

2 sin cos = cos

22

2

:

x cos

2 sin x - 1

=0

2

2

:

.2 sin

x 2

-1

=

0

,cos

x 2

= 0

x

= + k

2

2

x = + 2k

x 2 sin = 1

2

x

1

sin =

2

2

:

x

x 5

= + 2k or = + 2k

26

26

:

5

x = + 4k or x = + 4k

3

3

. + 2k

sin ( - x) = sin x cos ( + x) = - cos x ()

- cos x = sin x

. ,tan x = -1

4

Mathematics, Summer 2011 / Exercise 7 ? Notes

tan x + 4

()

: .2

1 + sin x ()

1 - sin2 x

1 1 - cos x ()

2x - 3 sin x ()

.R ()

.x = 2k ,cos x = 1 ()

,cos2 x > 0 x .1 - sin2 x cos2 x ()

.x

=

2

+

k

,cos x

=

0

.x =

4

+ k

.x +

4

=

2

+ k

()

.cot (x) =

cos x sin x

,

.3

?cot (x) () .x = k ,sin x = 0

. cot x ()

cos (x + ) - cos x cos x cot (x + ) = sin (x + ) = - sin x = sin x = cot x

. cot x

f (x) = 3 + 4 sin (2x) : .4 ? .-1 7

?3 sin (2x - 1) .5

3 sin (2x - 1) .1 -1 sin (2x - 1) , 2x - 1 .[-3, 3] : .3 -3

.sin x + cos x .6

, (!)

.x =

4

,

.

sin x = cos x

. sin x= cos x , . 2

.-

2,

2 , , .-

2

.x =

5 4

,

5

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