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Bài tập phương trình lượng giác | Môn toán cao cấp

53

Đường tròn lượng giác ▶ Là đường tròn có tâm O trong mặt phẳng Oxy và có bán kính R = 1. ▶ A(1;0) là điểm gốc của đường tròn lượng giác.Chiều dương là chiều ngược chiều kim đồng hồ. Chiều âm là chiều cùng chiều kim đồng hồ.Tài liệu giúp bạn tham khảo, ôn tập và đạt kết quả cao. Mời bạn đọc đón xem !

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Tác giả:

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Ma Thu Hương

6 tháng trước
Danh sách Quiz
/59
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
Chuyên Á 1
PH¯¡NG TRÌNH L¯þNG GIÁC
XÁC ĐÞNH GIÁ TRÞ L¯þNG GIÁC
lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
2
CĂA MÞT G C HAY CUNG L¯þNG GIÁC
✓
1
. Đ°ßng tròn l°ÿng giác
Là đ°ờng tròn có
t m
trong mặt phẳng
Oxy
v c bÆn k nh
1
R
.
ø
ù
0
1
;
A
là điểm gốc của đ°ờng tròn l°ÿng giác.
Chiều d°¡ng là chiều ng°ÿc chiều kim đồng hồ.
Chiều âm là chiều cùng chiều kim đồng hồ.
. Giá trß l°ÿng giác
2
Cho cung
AM
sin
OK
Tung độ của điểm
M
.
Cos =
cos
OH
Hoành độ của điểm
M
.
sin
tan
cos
AT
(KØo d i
OM
cắt tiếp tuyến
’
t At
tại
T
)
cos
cot
sin
B
(KØo d i
OM
cắt tiếp tuyến
’
s Bs
tại
S
, với
ø
ù
0
;
1
B
)
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
4
CÔNG THþC L¯þNG GIÁC ✓
1. Hß thÿc c¢ bÁn
1 tan2 a 12 tana 1
1
cos a
tan .cota a 1
cota
sin
2
a cos
2
a
1 cot2 a 12 cota 1
sin atana
tana
sina
sina tan cosa a cota
cosa
cosa cot .sina
a cosa sina
2. Cung liŒn k¿t
3. C ng thÿc cßng
sin(a b ) sin cosa b sin cosb a
cos(a b) cos cosa b sin sina
b
cos( a) cosa
sin( a) sina
tan( a) tana
cot( a) cota
sin( a) sina
cos( a) cosa
tan( a) tana
cot( a) cota
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
ö ö
sin÷ a÷ cosa ø 2
ø
ö ö
cos÷ a÷ sina ø
2 ø ö ö
tan÷ a÷ cota ø 2
ø
ö ö
cot÷ a÷ tana ø 2
ø
ö ö
sin÷ a÷ cosa ø 2
ø
ö ö
cos÷ a÷ sina ø
2 ø ö ö
tan÷ a÷ cota ø 2
ø
ö ö
cot÷ a÷ tana ø 2
ø
tan(a b ) tana tanb
1 tan .tana b
cot(a b )
`p dāng c ng thÿc cßng ta °ÿc
ö ö
sinx cosx 2 sin÷x ÷
ø
4 ø ö
ö
cosx sinx 2 cos÷x ÷
ø 4 ø
…
4. Cung liŒn k¿t
5. C ng thÿc nhân ôi
sin2a 2sin cosa a
cos
2
a–sin
2
a
cos2a 1 2sin
2
a
2cos
2
a 1
tan2a 2tana2
1 tan a
cot2a 1 1 tan2 a tan2a
2tana
sinøa ù sina
cosøa ù
cosa tanøa ù
tana cotøa ù
cota
sin( a) sina
cos( a) cosa
tan( a) tana
cot( a) cota
sin÷ a÷ cosa cosö÷ aö÷ sina ö ö
ø 2 ø ø 2 ø
tan÷ a÷ cota cotö÷ aö÷ tana ö ö
ø 2 ø ø 2 ø
sinøa k .2 ù sina cosøa k .2 ù cosa
tanøa k ù tana cotøa k ù cota
6.C ng thÿc h¿ b¿c
*sin
2
a
1
cos2a
1cos2a 2sin
2
a
2
*cos
2
a
1
cos2a
1cos2a 2cos
2
a
2
*tan2 a 1
cos2a 1
cos2a
lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
6
7. C ng thÿc bi¿n t ch th nh táng *cos
.cosa b ù ûcos(a b ) cos(a b ù)û
b
ù
1
ûcos(a b )
*sin .sina
cos(a b ù)û
*sin .cosa b ùûsin(a b ) sin(a
b ù)û 2
*cos .sina b
ù
1
ûsin(a b )
sin(a b ù)û 2
8.C ng thÿc bi¿n táng th nh t ch
*cosa cosb 2cos
÷
a b
ö
÷
cosö
÷
a b
ö
÷
ö
ø 2 ø ø 2 ø
cosa cosb 2sinö
÷
a b
ö
÷
sinö
÷
a b
ö
÷
ø 2 ø ø 2 ø
ö
*sina sinb 2sin
÷
a b
ö
÷
cosö
÷
a b
ö
÷
ø 2 ø ø 2 ø
sina sinb 2cosö÷a b ö÷sinö÷a b ö÷
ø 2 ø ø 2 ø
*tana tanb sin(a
b ) cos cosa
b
Ghi chØp: ........................................................................................................................
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VÂN ĐÀ 1
PH¯¡NG TRÌNH L¯þNG GIÁC C¡ BÀN ✓
A. PH¯¡NG PH`P
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
1.
hoặc
æo
ùu
v k 360
0
2. hoặc cosu cosv ú 0 ,k úûu v k360
3.
4.
æo
ùu v k 360
0
sinu sinv ú 0 0 ,k
úûu 180 v k360
radian ù u v k2
sinu sinv úû v k2 ,k
u
radian ù u v k2
cosu cosv úû v k2 ,k
u
radian tanu tanv
u v k , k
hoặc
æo tanu tanv u v
k 180
0
, k
radian cotu cotv
u v k , k
hoặc
æo cotu cotv u v k
180
0
, k
radian
ù x arcsinm k 2
sinx m úû arcsinm k 2 ,k
x
ĐẶC
BIÞT
sinu
1 u
k2
2
sinu
1
u k2
2
sinu
0 u k
cosu
1 u k2
cosu
1 u k2
radian
ù x arccosm k 2
cosx m úû arccosm k 2 ,k
x
radian
tanx m x arctanm k ,
k
radian
cotx m x arccotm k ,
k
lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
8
5.
6.
7.
8.
￧ Ghi chép bá sung
..................................................................................................................................................................
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cosu
0
u k
2
B. VÍ DĀ
ö ö
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
Ví dā 1. Giải ph°¡ng trình sin÷x
÷ sin
2
.
ø 2 ø 5
Lßi giÁi
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Ví dā 2. Giải ph°¡ng trình sinøx 30
0
ù sin20
0
.
Lßi giÁi
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Ví dā 3. Giải ph°¡ng trình sinxøcosx 2ù 0
Lßi giÁi
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Ví dā 4. Giải ph°¡ng trình 2sinx 1.
Lßi giÁi
lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
10
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ö ö
Ví dā 5. Giải ph°¡ng trình sin÷x
÷ 1.
ø 5 ø
Lßi giÁi
..................................................................................................................................................................
.....
.......................................................................................................................................................................
.......................................................................................................................................................................
.......................................................................................................................................................................
ö ö
Ví dā 6. Giải ph°¡ng trình 2sin÷x
÷ 2 0 .
ø 6 ø
Lßi giÁi
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Ví dā 7. Giải ph°¡ng trình 2sinøx 60
0
ù 3 0 .
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
Lßi giÁi
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Ví dā 8. Giải ph°¡ng trình sin3x cos÷öø
x 3 ö
÷
ø .
Lßi giÁi
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Ví dā 9.
Giải ph°¡ng trình cos 2÷öø
x ö÷ø
sin÷øö
x 4
öø÷ .
5
Lßi giÁi
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Ví dā 10. Giải ph°¡ng trình 5sinx 1 .
lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
12
Lßi giÁi
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Ví dā 11. Giải ph°¡ng trình sinøx 2ù .
Lßi giÁi
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Ví dā 12. Giải ph°¡ng trình cos÷x
2
ö÷ cos .
ö
ø 5 ø 7
Lßi giÁi
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Ví dā 13. Giải ph°¡ng trình cosøx 40
0
ù cos10
0
.
Lßi giÁi
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
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Ví dā 14. Giải ph°¡ng trình 2cosx 1 .
Lßi giÁi
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ö ö
Ví dā 15. Giải ph°¡ng trình cos÷x
÷ 1 .
ø 8 ø
Lßi giÁi
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ö ö
Ví dā 16. Giải ph°¡ng tr nh 2cos÷x
÷ 3 0 .
ø 6 ø
Lßi giÁi
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lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
14
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Ví dā 17. Giải ph°¡ng trình 2cosøx 30
0
ù 2 0 .
Lßi giÁi
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Ví dā 18. Giải ph°¡ng trình 3cosx 3 0 .
Lßi giÁi
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Ví dā 19. Giải ph°¡ng trình 5cosø2x 1ù 30 .
Lßi giÁi
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lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
Ví dā 20. Giải ph°¡ng trình tan÷x
3
ö÷ tan .
ö
ø 4 ø 7
Lßi giÁi
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Ví dā 21. Giải ph°¡ng trình tanøx 48
0
ù tan25
0
.
Lßi giÁi
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Ví dā 22. Giải ph°¡ng trình tanx 3 .
Lßi giÁi
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ö ö
Ví dā 23. Giải ph°¡ng trình tan÷ x÷ cot .
ø 6 ø 6
lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
16
Lßi giÁi
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Ví dā 24. Giải ph°¡ng trình tanøx 30
0
ù 10.
Lßi giÁi
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Ví dā 25. Giải ph°¡ng trình 2tanx 5 0.
Lßi giÁi
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ö ö
Ví dā 26. Giải ph°¡ng trình cot÷x
÷ cot .
ø 8 ø 5
Lßi giÁi
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lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
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Ví dā 27. Giải ph°¡ng trình cotøx 28
0
ù cot12
0
.
Lßi giÁi
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Ví dā 28. Giải ph°¡ng trình 3cotx 3 0.
Lßi giÁi
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ö ö
Ví dā 29. Giải ph°¡ng trình cot÷x
÷ 1 0 .
ø 6 ø
Lßi giÁi
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lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
18
Ví dā 30. Giải ph°¡ng trình cot 2ø x 20
o
ù 3 0.
Lßi giÁi
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Ví dā 31. Giải ph°¡ng trình 3cotx 5 0 .
Lßi giÁi
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Ví dā 32. Giải ph°¡ng trình cosx sinx 0.
Lßi giÁi
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Ví dā 33. Giải ph°¡ng trình 3sinx cosx 0.
Lßi giÁi
lOMoARcPSD| 49519085
Tr°ờng THPT MARIE CURIE
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Ví dā 34. Giải ph°¡ng trình 3cosx sinx 0.
Lßi giÁi
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Ví dā 35. Giải ph°¡ng trình sin2x sin2 cosx x 0.
Lßi giÁi
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Ví dā 36. Giải ph°¡ng trình cosx 2sin
2
x
0 .
2
Lßi giÁi
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lOMoARcPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
20
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Ví dā 37. Giải ph°¡ng trình sin3x sinx 0 .
Lßi giÁi
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Ví dā 38. Giải ph°¡ng trình cos4x sinx .
Lßi giÁi
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ù
Ví dā 39. Ph°¡ng trình 2cos
2
x 1 0 có bao nhiêu nghiệm trên đoạn
ú
3
;
ù
ú .
û 2 2 û
Lßi giÁi
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lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE Chuyên Á 1
PH¯¡NG TRÌNH L¯þNG GIÁC
XÁC ĐÞNH GIÁ TRÞ L¯þNG GIÁC lOMoAR cPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC CĂA MÞT G C HAY CUNG L¯þNG GIÁC ✓
1 . Đ°ßng tròn l°ÿng giác
 Là đ°ờng tròn có t m O trong mặt phẳng Oxy v c bÆn k nh R 1 .  Aø 1 ;0 ù
là điểm gốc của đ°ờng tròn l°ÿng giác.
 Chiều d°¡ng là chiều ng°ÿc chiều kim đồng hồ.
 Chiều âm là chiều cùng chiều kim đồng hồ. 2. G
iá trß l°ÿng giác Cho cung AM  sin OK
Tung độ của điểm M .  Cos = cos OH
Hoành độ của điểm M .  sin tan AT t At tại T ) c
(KØo d i OM cắt tiếp tuyến ’ os  cos cot 0 ; 1 ) s
(KØo d i OM cắt tiếp tuyến ’
s Bs tại S , với ø ù in B B 2 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE lOMoAR cPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
CÔNG THþC L¯þNG GIÁC ✓ 1. Hß thÿc c¢ bÁn 1 tan2 a 12 tana 1 sin2 a cos2 a 1cos a tan .cota a 1cota 1 cot2 a 12 cota 1 sin atana tana
sina sina tan cosa a cota
cosa cosa cot .sina
a cosa sina 2. Cung liŒn k¿t 3. C ng thÿc cßng
sin(a b ) sin cosa b sin cosb a cos( a) cosa sin( a) sina sin( a) sina
cos(a b) cos cosa b sin sina cos( a) cosa tan( a) tana b tan( a) tana cot( a) cota cot( a) cota 4 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE ö ö ö ö
tan(a b ) tana tanb
sin÷ a÷ cosa ø 2
sin÷ a÷ cosa ø 2 1 tan .tana b ø ø ö ö ö ö cot(a b )
cos÷ a÷ sina ø cos÷ a÷ sina ø 2 ø ö ö 2 ø ö ö
tan÷ a÷ cota ø 2
`p dāng c ng thÿc cßng ta °ÿc tan÷ a÷ cota ø 2 ø ø ö ö ö ö ö ö
sinx cosx 2 sin÷x ÷
cot÷ a÷ tana ø 2 ø cot÷ a÷ tana ø 2 ø 4 ø ö ø ö
cosx sinx 2 cos÷x ÷ ø 4 ø … 4. Cung liŒn k¿t
5. C ng thÿc nhân ôi
sin2a 2sin cosa a sin( a) sina sinøa ù sina cos2 a–sin2 a cos( a) cosa cosøa ù tan( a) tana
cos2a 1 2sin2 a cot( a) cota cosa tanøa ù 2cos2 a 1 tana cotøa ù tan2a 2tana2 cota 1 tan a
sin÷ a÷ cosa cosö÷
aö÷ sina ö ö cot2a 1 1 tan2 a tan2a ø 2 ø ø 2 ø 2tana
tan÷ a÷ cota cotö÷
aö÷ tana ö ö ø 2 ø ø 2 ø 6.C ng thÿc h¿ b¿c
sinøa k .2 ù sina cosøa k .2 ù cosa *sin2 a
1 cos2a 1cos2a 2sin2 a 2
tanøa k ù tana cotøa k ù cota *cos2 a
1 cos2a 1cos2a 2cos2 a 2 *tan2 a 1 cos2a 1 cos2a lOMoAR cPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
7. C ng thÿc bi¿n t ch th nh táng *cos
8.C ng thÿc bi¿n táng th nh t ch
.cosa b ù ûcos(a b ) cos(a b ù)û
*cosa cosb 2cos÷a b ö÷cosö÷a b ù ö÷ ö *sin .sina b 1 ûcos(a b ) ø 2 ø ø 2 ø cos(a b ù)û cosa cosb
2sinö÷a b ö÷sinö÷a b *sin .cosa
b ùûsin(a b ) sin(a b ù)û 2 ö÷ ø 2 ø ø 2 ø ù *cos .sina b 1 ûsin(a b ) ö sin(a b ù)û 2
*sina sinb 2sin÷a b ö÷cosö÷a b ö÷ ø 2 ø ø 2 ø
sina sinb 2cosö÷a b ö÷sinö÷a b ö÷ ø 2 ø ø 2 ø *tana tanb sin(a b ) cos cosa b
Ghi chØp: ........................................................................................................................
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........................................................................................................................................... VÂN ĐÀ 1
PH¯¡NG TRÌNH L¯þNG GIÁC C¡ BÀN ✓ A. PH¯¡NG PH`P 6 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE  1. radian ù u v k2 æo ùu v k 3600 hoặc sinu sinv ú 0 0 ,k
sinu sinv úû v k2 ,k
úûu 180 v k360 æo ùu u radian ù u v k2
cosu cosv úû v k2 ,k u v k 3600
2. hoặc cosu cosv ú 0 ,k úûu v k360
radian tanu tanv
æo tanu tanv u v
3. u v k , k
hoặc k 1800, k
radian cotu cotv
æo cotu cotv u v k
4. u v k , k hoặc 1800, k radian
ù x arcsinm k 2 sinx m úû
arcsinm k 2 ,k ĐẶC x BIÞT radian  sinu
ù x arccosm k 2 1 u cosx m úû
arccosm k 2 ,k k2 x 2 radian  sinu
tanx m x arctanm k , k 1 radian u k2
cotx m x arccotm k , k 2  sinu 0 u k  cosu 1 u k2  cosu 1 u k2 lOMoAR cPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC  cosu 5. 0 u k 2  6. 7. 8.
￧ Ghi chép bá sung
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.................................................................................................................................................................. B. VÍ DĀ ö ö 8 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE
Ví dā 1. Giải ph°¡ng trình sin÷x ÷ sin 2 . ø 2 ø 5 Lßi giÁi
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Ví dā 2. Giải ph°¡ng trình sinøx 30 ù 0 sin200 . Lßi giÁi
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Ví dā 3. Giải ph°¡ng trình sinxøcosx 2ù 0 Lßi giÁi
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Ví dā 4. Giải ph°¡ng trình 2sinx 1. Lßi giÁi lOMoAR cPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC
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Ví dā 5. Giải ph°¡ng trình sin÷x ÷ 1. ø 5 ø Lßi giÁi
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Ví dā 6. Giải ph°¡ng trình 2sin÷x ÷ 2 0 . ø 6 ø Lßi giÁi
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Ví dā 7. Giải ph°¡ng trình 2sinøx 60 ù 0 3 0 . 10 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE Lßi giÁi
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Ví dā 8. Giải ph°¡ng trình sin3x cos÷öø x 3 ö÷ø . Lßi giÁi
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Ví dā 9. Giải ph°¡ng trình cos 2÷öø x ö÷ø sin÷øö x 4 öø÷ . 5 Lßi giÁi
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Ví dā 10. Giải ph°¡ng trình 5sinx 1 . lOMoAR cPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC Lßi giÁi
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Ví dā 11. Giải ph°¡ng trình sinøx 2ù . Lßi giÁi
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Ví dā 12. Giải ph°¡ng trình cos÷x 2 ö÷ cos . ö ø 5 ø 7 Lßi giÁi
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Ví dā 13. Giải ph°¡ng trình cosøx 40 ù 0 cos100 . Lßi giÁi 12 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE
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Ví dā 14. Giải ph°¡ng trình 2cosx 1 . Lßi giÁi
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Ví dā 15. Giải ph°¡ng trình cos÷x ÷ 1 . ø 8 ø Lßi giÁi
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Ví dā 16. Giải ph°¡ng tr nh 2cos÷x ÷ 3 0 . ø 6 ø Lßi giÁi
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ChuyŒn đề 1: L¯þNG GIÁC
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Ví dā 17. Giải ph°¡ng trình 2cosøx 30 ù 0 2 0 . Lßi giÁi
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Ví dā 18. Giải ph°¡ng trình 3cosx 3 0 . Lßi giÁi
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Ví dā 19. Giải ph°¡ng trình 5cosø2x 1ù 30 . Lßi giÁi
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....................................................................................................................................................................... 14 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE
Ví dā 20. Giải ph°¡ng trình tan÷x 3 ö÷ tan . ö ø 4 ø 7 Lßi giÁi
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Ví dā 21. Giải ph°¡ng trình tanøx 48 ù 0 tan250 . Lßi giÁi
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Ví dā 22. Giải ph°¡ng trình tanx 3 . Lßi giÁi
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Ví dā 23. Giải ph°¡ng trình tan÷ x÷ cot . ø 6 ø 6 lOMoAR cPSD| 49519085
ChuyŒn đề 1: L¯þNG GIÁC Lßi giÁi
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Ví dā 24. Giải ph°¡ng trình tanøx 30 ù 0 10. Lßi giÁi
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Ví dā 25. Giải ph°¡ng trình 2tanx 5 0. Lßi giÁi
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Ví dā 26. Giải ph°¡ng trình cot÷x ÷ cot . ø 8 ø 5 Lßi giÁi
....................................................................................................................................................................... 16 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE
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Ví dā 27. Giải ph°¡ng trình cotøx 28 ù 0 cot120 . Lßi giÁi
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Ví dā 28. Giải ph°¡ng trình 3cotx 3 0. Lßi giÁi
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Ví dā 29. Giải ph°¡ng trình cot÷x ÷ 1 0 . ø 6 ø Lßi giÁi
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ChuyŒn đề 1: L¯þNG GIÁC
Ví dā 30. Giải ph°¡ng trình cot 2ø x 20 ù o 3 0. Lßi giÁi
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Ví dā 31. Giải ph°¡ng trình 3cotx 5 0 . Lßi giÁi
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Ví dā 32. Giải ph°¡ng trình cosx sinx 0. Lßi giÁi
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Ví dā 33. Giải ph°¡ng trình 3sinx cosx 0. Lßi giÁi 18 lOMoAR cPSD| 49519085 Tr°ờng THPT MARIE CURIE
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Ví dā 34. Giải ph°¡ng trình 3cosx sinx 0. Lßi giÁi
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Ví dā 35. Giải ph°¡ng trình sin2x sin2 cosx x 0. Lßi giÁi
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Ví dā 36. Giải ph°¡ng trình cosx 2sin2 x 0 . 2 Lßi giÁi
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ChuyŒn đề 1: L¯þNG GIÁC
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Ví dā 37. Giải ph°¡ng trình sin3x sinx 0 . Lßi giÁi
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Ví dā 38. Giải ph°¡ng trình cos4x sinx . Lßi giÁi
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Ví dā 39. Ph°¡ng trình 2cos2 x 1 0 có bao nhiêu nghiệm trên đoạn ú 3 ; ú . û 2 2 û Lßi giÁi
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