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
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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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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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