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
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