Trigonometrikus egyenletek

x2

=

5 6

+k

? 2

x1

=

6

+k

? 2

Trigonometrikus egyenletek

1. Oldd meg a val?s sz?mok halmaz?n a k?vetkez egyenleteket!

2

a) sin x = 2

b)

sin x

=

1 2

x1 = 4 + k ? 2

3 x2 = 4 + k ? 2

c) cos x = -1

d) cos x = -0, 5

x = (2k + 1) ?

x2 = 3 + k ? 2

e)

sin x = -

3 2

4

2 x = ? + k ? 2

3

x1

=

- 3

+

k

?

2

x1 = 12 + k ?

5 x2 = 12 + k ?

x2

=

3

+k

?

x1

=

6

+k

?

24

2

2. Oldd meg a val?s sz?mok halmaz?n a k?vetkez egyenleteket!

a) sin 3x = 0

b)

sin 2x

=

1 2

x=k?

3

1 c) cos 2x =

2

?

k

+

6

-

=

x2

3

d) sin 2x = 2

x1

=

6

+

k

?

- 9

3

1

e) cos 3x - = -

3

2

2

?

k

+

=

x2

2 f) sin 4x - = 1

3

x1 = 3 + k ? 3

2

2

g) sin 5x - = -

4

2

2

x2 = 10 + k ? 5

3

h) tg x - = 3

6

x1 = k ? 5

2

i) 2 cos 3x = 1

2

3

?

k

+

- 12

=

x2

2

j) sin x = 2

x= +k?

7

x

=

2

+k

?

1 x1 = 4 + k ? 2

3 x2 = 4 + k ? 2

x1 = 12 + k ? 3

2

x2

=

19

+k?

2 19

x1

=

k

?

2 7

,

3. Oldd meg az al?bbi egyenleteket! a) sin 8x = sin 11x

c) sin 4x = sin 9x

e) sin 10x = sin 15x

2 13

2 19

?

?

?

k

,

k

,

k

+

2 5

+

2 3

+

?

k

13

?

k

19

=

=

=

=

=

x2

x1

x2

x1

x2

b) sin 6x = sin 13x d) cos 9x = cos 15x f) cos 5x = cos 8x

x2

=

k

?

12

x1

=

k

?

3

,

x2

=

k

?

2 13

25

2 25

x1

=

k

?

2 3

,

x1

=

k

?

2 5

,

2019.02.25. 17:27 (rev2602 )

1. oldal

Udvari Zsolt ? uzsolt.hu

Udvari Zsolt ? uzsolt.hu

x1 x2

= =

-122+4 +k

?

2

k?

4

x1

=

12

+k?

2

x2

=

33

+k?

11

x1

=

8

+k?

1

x2

=

3 56

+k?

7

x1 x2

= =

3428++kk?

2

?

12 7

x1

=

3

+k?

2 1

x2

=

2 33

+k?

2 11

x1

=

20

+k?

2

x2

=

-

40

+

k

?

4

x1

=

7

+k?

2 1

x2

=

2 21

+k?

2 9

x1

=

2

+k?

2 1

x2

=

22

+k?

2 11

x1

=

k

?

2 5

,

x2

=

k

?

2 27

x1

=

k

?

2 9

,

x2

=

19

+k?

2 19

x1 x2

= =

k?

15

2 5

,

+k

?

2 15

x1

=

k

?

4

,

x2

=

k

?

15

x1

=

k

?

2 7

,

x2

=

27

+k?

2 27

x1

=

k

?

2 7

,

x2

=

17

+k?

2 17

x1

=

k

?

2 3

,

x2

=

k

?

2 25

h) cos 11x = cos 14x j) sin 5x = sin 12x l) sin 10x = sin 17x n) cos 11x = cos 19x p) sin 5x = sin 10x r) sin 5x = sin 14x t) cos 11x = cos 16x

b) sin 5x + = sin (6x)

2

d) sin 4x + = sin (5x)

7

f) cos 2x + = cos (6x)

5

h) sin 5x + = sin (6x)

3

j) sin 3x + = sin (4x)

4

l) sin 6x + = sin (8x)

4

n) sin 9x + = sin (13x)

3

p) cos 2x + = cos (6x)

3

2. oldal

x1 x2

= =

-163+2 +k

?

2

k?

4

x1

=

4

+k?

2 1

x2

=

3 44

+k

?

2 11

x1

=

6

+k?

2 3

x2

=

30

+k?

2 15

x1 x2

= =

8 20

++kk??25

x1

=

8

+k?

1

x2

=

3 40

+k

?

5

x1

=

16

+k?

1

x2

=

-

112

+

k

?

7

x1

=

15

+k?

2 3

x2

=

-

95

+

k

?

2 19

x1

=

9

+k?

2 3

x2

=

2 21

+k

?

2 7

x1

=

k

?

2 5

,

x2

=

k

?

2 17

x1

=

k

?

4

,

x2

=

22

+k?

11

x1 x2

= =

k k

? ?

2 27

,

11

x1

=

k

?

2 5

,

x2

=

k

?

2 25

x1

=

k

?

2 9

,

x2

=

k

?

2 21

x1 x2

= =

k?

22

2

,

+k

?

11

x1

=

k

?

3

,

x2

=

k

?

11

g) cos 8x = cos 14x

i) sin 9x = sin 13x

k) cos 6x = cos 15x

m) cos 10x = cos 15x

o) cos 2x = cos 9x

q) sin 7x = sin 15x

s) cos 6x = cos 11x

4. Oldd meg az al?bbi egyenleteket!

a) sin 2x + = sin (5x) 3

c) cos 8x + = cos (11x)

5

e) cos 6x + = cos (8x)

8

g) sin 4x + = sin (6x)

4

i) sin 3x + = sin (7x)

2

k) sin 6x + = sin (9x)

2

m) sin 5x + = sin (6x)

4

o) cos 2x + = cos (6x)

4

2019.02.25. 17:27 (rev2602 )

Udvari Zsolt ? uzsolt.hu

x1

=

4

+k

?

x2 = + k ?

x1

=

+ 24

k

?

3

x2

=

- 16

+

k

?

2

x=- +k?

6 x = ? k ? 2

x1

=

24

+k?

6

x2

=

4

+k?

1

x1

=

20

+k?

5

x2

=

-

4

+k

?

1

x1

=

18

+k?

2 9

x2

=

2

+k?

2 1

x1

=

24

+k?

6

x2

=

8

+k?

2

x1

=

18

+k?

2 9

x2

=

10

+k?

2 5

x1

=

24

+k?

6

x2

=

-

4

+k

?

1

x1

=

18

+k?

2 9

x2

=

-

2

+k

?

2 1

x1

=

18

+k?

2 9

x2

=

6

+k?

2 3

x1

=

28

+k?

7

x2

=

-

8

+k

?

2

b) cos 5x = sin 9x d) sin 3x = cos 6x f) cos 4x = sin 5x h) cos 5x = sin 7x j) sin 2x = cos 7x l) sin 4x = cos 8x n) sin 4x = cos 5x p) cos 4x = sin 6x r) sin 5x = cos 7x

3 3

b) cos x2 = 1

-

f) cos 5x = cos x -

4

3 h) -2 sin 2x - = 2

4

=

tg x

d)

3. oldal

x1 x2

= =

3

-

+

12

k

?

2

+k?

2

x

=

k

?

4

x

=

2

+k

?

x

=

10

+k

?

x

=

-

4

+

k

?

x

=

?

3

+

k

?

x1

=

30

+k?

2 15

x2

=

-

6

+k

?

2 3

x1

=

18

+k?

2 9

x2

=

10

+k?

2 5

x1

=

22

+k?

2 11

x2

=

-

6

+k

?

2 3

x1

=

20

+k?

5

x2

=

-

4

+k

?

1

x1

=

28

+k?

7

x2

=

4

+k?

1

x1

=

34

+k?

2 17

x2

=

-

10

+

k

?

2 5

x1

=

30

+k?

2 15

x2

=

6

+k?

2 3

x1

=

16

+k?

4

x2

=

-

4

+k

?

1

x1

=

22

+k?

2 11

x2

=

6

+k?

2 3

5. Oldd meg az al?bbi egyenleteket! a) sin 4x = cos 7x c) cos 3x = sin 5x e) sin 6x = cos 9x g) cos 6x = sin 11x i) sin 6x = cos 8x k) cos 4x = sin 6x m) cos 4x = sin 7x o) sin 2x = cos 7x q) cos 6x = sin 9x

6. Oldd meg az al?bbi egyenleteket!

3 4

=

sin2 x

c) tg x = -1

e) tg 5x = tg x

g) 2 ? cos 4x - = - 3

2

2019.02.25. 17:27 (rev2602 )

a)

2 x = ? + k ? 2

3

x1 = 6 + k ? 2

5 x2 = 6 + k ? 2

x1 = 6 + k ? 2

5 x2 = 6 + k ? 2

7. Oldd meg a val?s sz?mok halmaz?n a k?vetkez, m?sodfok?ra visszavezethet egyenleteket!

a) sin2 x - 2 sin x = 0

b) 2 sin2 x - 7 sin x + 3 = 0

x=k?

c) 8 sin2 x - 7 cos2 x = 8 e) sin2 x - cos2 x = cos x

2

?

k

+

? 3

x= +k?

2

=

x2

d) cos2 x - sin2 x = 1 2

f) 2 sin2 x + 3 cos x = 0

x1 = + k ? 2

x1 = k ? 3

x2 = 2 + k ? 2

g) cos2 x - sin x = 1

h) -4 cos2 x - 2 sin x + 3 = 0

i) sin2 x + cos x = 1 k) 4 cos2 x+17 sin x = 8

j) 2 sin2 x + 5 ? cos x - 4 = 0 l) tg2 x = 2 tg x

x4

=

11 10

+ k ? 2

x3

=

-

10

+

k

?

2,

x2

=

7 10

+k

? 2,

x1

=

3 10

+k

? 2,

x2

=

-

3

+

k

?

2

x1

=

3

+k

? 2

x1 = 1429 + k ? 360 x2 = 16531 + (2k + 1) ? 360

x1 = 2 + k ? x2 = 0 + k ? 2

x1 = k ? x2 = 63,43 + k ? 180

2019.02.25. 17:27 (rev2602 )

4. oldal

Udvari Zsolt ? uzsolt.hu

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download