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cong thuc toan hoc luong giac

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  • 1. L­îng gi¸c I. B¶NG L¦îNG GI¸C 0 Sin  6 1 2 4 2 2 2 2 3 3 2 1 2 2 1 3 1 3 3 0 Cot  1 Tan  0 Cos  || 3 2 3 3 3 2 3 3 2 1  2 3 4 2 2 2  2 ||  3 -1 0  3 3 -1 1 0 II. §¼ng thøc, BÊt ®¼ng thøc hay gÆp: 1  , x ≠  k  ( k  Z) 2 cos x 2 1 2. 1 + cot 2 x  , x ≠ k (k Z) sin 2 x 3 3. cos3 x sin 3 x  cos 3 x sin 3 x  sin 4 x 4 3 3 4. cos x cos 3x  sin x sin 3x  cos3 2 x 1  cos 2 2 x 3  cos 4 x 5. cos 4 x  sin 4 x   2 4 2 1  3cos 2 x 5  3cos 4 x 6. cos 6 x  sin 6 x   4 8 A 7.  sin A  4 cos 2 1. 1 + tan 2 x  8. sin 2 A  4 sin A 9. sin 3 A  4 cos 3A 2 10. sin 4 A  4 sin 2 A 11. cos A  1  4 sin A 2 12. cos 2 A  1  4 cos A 13. cos 3 A  1  4 sin 3A 2 14. cos 4 A  1  4 cos 2 A 15. tan A   tan A 16. tan 2 A   tan 2 A 17. cot A cot B  1 A B tan  1 2 2 A A 19. cot   cot 2 2 18. tan 3 3 2 A 3 1   sin  2 2 3 A   sin 2  1 4 2 3 3  sin A  8 A 1  sin 2  8 9  sin 2 A  4 3 1   cos A  2 A 3 3 2   cos  2 2 A 9 2   cos 2  2 4 1  cos A  8 A 3 3  cos 2  8  sin A   tan A  3 A  tan 2  3 3 A 1 2 1 A  tan 2  3 3  tan 2 A  9  tan 2 5 6 1 2 3 2 3  3   3  0 -1 0 ||
  • 2. III, quan hÖ gi÷a c¸c gi¸ trÞ l­îng gi¸c Sin Cos Tan Cot x  k -x  x Tan x Cot x x  k 2 - sin x Cos x - tan x - cot x Sin x - cos x - tan x - cot x Sin x Cos x  x x x 2 Cos x Sin x Cot x Tan x 2 Cosx - sin x - cot x - tan x - Sin x - cos x - tan x Cot x IV, C«ng thøc biÕn ®æi 1. C«ng thøc c«ng: sin(a  b)  sin a cos b  cos a sin b 5. C«ng thøc tæng -> tÝch cos(a  b)  cos a cos b  sin a sin b ab a b cos 2 2 ab a b sin a  sin b  2 cos sin 2 2 ab a b cos a  cos b  2 cos cos 2 2 ab a b cos a  cos b  2sin sin 2 2 sin( a  b) tan a  tan b  cos a cos b sin(b  a ) cot a  cot b  sin a sin b 2 tan a  cot a  sin 2a cot a  tan a  2cot 2a cos(a  b)  cos a cos b  sin a sin b tan a  tan b 1  tan a tan b tan a  tan b tan(a  b)  1  tan a tan b tan(a  b)  2. C«ng thøc nh©n ®«i, nh©n ba: sin 2a  2sin a cos a cos 2a  cos 2 a  sin 2 a  2cos 2 a  1 2 tan a tan 2 a  1  tan 2 a sin 3a  3sin a  4sin 3 a cos3a  4 cos3 a  3cos a tan 3a  3 tan a  tan 3 a 1  tan 2 a 3. C«ng thøc h¹ bËc: 1  cos 2 a 2 1  cos 2a cos 2 a  2 3sin a  sin 3a sin 3 a  4 3cos a  cos 3a cos3 a  4 sin 2 a  4. C«ng thøc tÝch -> tæng sin(a  b )  sin(a  b) 2 sin(a  b )  sin(a  b) cos a sin b  2 cos( a  b)  cos(a  b) cos a cos b  2 cos(a  b)  cos( s  b ) sin a sin b  2 sin a cos b  sin a  sin b  2sin * §Æc biÖt:    sin a  cos a  2 sin  a    2 cos( a  ) 4 4     sin a  cos a  2 sin(a  )   2 cos  a   4 4    sin a  3 cos a  2sin(a  )  2 cos( a  ) 3 6   sin a  3 cos a  2sin( a  )  2 cos( a  ) 3 6   3 sin a  cos a  2sin(a  )  2 cos( a  ) 6 3    3 sin a  cos a  2sin( a  )  2 cos( a  ) 6 3 a 6. BiÓu diÔn qua t  tan 2 2t sin a  1 t2 1 t2 cos a  1 t 2 2t tan a  1 t 2
  • 3. V. HÖ thøc L­îng trong tam gi¸c: 1. §Þnh lý Sin: a b c    2R sin A sin B sin C 2 2 2 2. §Þnh lý Cosin: a  b  c  2bc cos A -> HÖ qu¶: cos A  b2  c 2  a 2 2bc 3. §Þnh lý h×nh chiÕu: a  b cos C  c cos B; b  c cos A  a cos C ; c  a cos B  b cos A a  r (cot B C A C A B  cot ); b  r (cot  cot ); c  r (cot  cot ) 2 2 2 2 2 2 4. §Þnh lý Cotang: cot A  R(b 2  c 2  a 2 ) (b 2  c 2  a 2 )  abc 4S 5. C«ng thøc ®­êng trung tuyÕn: 2 ma  b2  c 2 a 2  2 4 6. C«ng thøc ph©n gi¸c: A 2  2 bcp ( p  a ) bc bc 2bc cos la  7. C«ng thøc diÖn tÝch: 1 1 abc aha  ab sin C  2 R 2 sin A sin B sin C   pr  ( p  a )ra  2 2 4R S p ( p  a )( p  b)( p  c) 8. §é dµi c¸c b¸n kÝnh: abc S S A A ; r  ; ra  ; r  ( p  a ) tan ; ra  p tan 4S 2 2 p pa A A B C r  4 R sin ; ra  4 R sin cos cos 2 2 2 2 9. Trong tam gi¸c ABC cã: sin( B  C )  sin A; tan( B  C )   tan A; cos( B  C )   cos A;cot( B  C )   cot A R sin BC A BC A BC A BC A  cos ; tan  cot ;cos  sin ;cot  tan 2 2 2 2 2 2 2 2 VI, Ph­¬ng tr×nh l­îng gi¸c c¬ b¶n:    1  x   2  k 2  sin x  0  x  k   1  x   k 2 2   1  x    k 2   cos   0  x   k 2  1  x  k 2  ( k Z )     1  x   4  k  tan x   0  x  k   1  x   k 4      1  x   4  k   cot x  0  x   k  2   1  x   k  4  ( k Z )

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