Phơng trình lợng giác góc lợng giác & công thức lợng giác i.lý thuyết 1.giá trị l ơng giác của góc l ợng giác a.các định nghĩa : sin = OK cos = OH tan = AT cot = BU b. tính chất i> sin ( + k2 ) = sin cos ( + k2 ) = cos ; k Z tan ( + k ) = tan cot ( + k ) = cot ; k Z ii> với ta có : - 1 sin 1 ; - 1 cos 1 iii> cos 2 + sin 2 = 1 tan .cot = 1 1 + tan 2 = 2 cos 1 ( cos 0 ) 1 + cot 2 = 2 sin 1 ( sin 0 ) c. dấu các hàm số l ợng giác : d. bảng hàm số của cung l ợng giác đặc biệt Chú ý : + > sin = 0 = k ; k Z Góc phần t Số đo của góc sin cos tan cot I 0 < < /2 + + + + II /2 < < + - - - III < < 3 /2 - - + + IV 3 /2 < < 2 - + - - + > sin = 1 = /2 + k2 ; k Z +> sin = - 1 = - /2 + k2 ; k Z + > cos = 0 = /2 + k ; k Z +> cos = 1 = k2 ; k Z +> cos = - 1 = + k2 ; k Z 2. giá trị l ơng giác của các góc có liên quan đặc biệt i> cung đối nhau : cos ( - ) = cos sin ( - ) = - sin tan ( - ) = - tan cot ( - ) = - cot ii> cung hơn kém : sin ( + ) = - sin cos( + ) = - cos tan( + ) = tan cot( + ) = cot iii> cung bù nhau : sin ( - ) = sin cos ( - ) = - cos tan( - ) = - tan cot( - ) = - cot iv> cung phụ nhau : sin ( /2 - ) = cos cos ( /2 - ) = sin tan ( /2 - ) = cot cot( /2 - ) = tan v> cung hơn kém /2 : sin ( /2 + ) = cos cos ( /2 + ) = - sin tan ( /2 + ) = - cot cot( /2 + ) = - cot 3. công thức l ợng giác a. công thức cộng : cos( x y ) = cosx.cosy + sinx.siny ( 1) cos( x + y ) = cosx.cosy sinx.siny ( 2 ) sin( x y ) = sinx.cosy cosx.siny ( 3) sin( x + y) = sinx.cosy + cosx.siny ( 4 ) tan( x y ) = yx yx tan.tan1 tantan + ( 5 ) tan( x + y ) = yx yx tan.tan1 tantan + ( 6 ) b. công thức nhân đôi : i> công thức nhân đôi : sin 2x = 2sinx.cosx ( 7) công thức nhân 3 : cos 2x = cos 2 x sin 2 x ( 8 ) sin3x = 3sinx 4sin 3 x tan 2x = x x 2 tan1 tan2 ( 9 ) cos3x = 4cos 3 x 3cosx ii> công thức hạ bậc : sin 2 x = 2 2cos1 x ( 10 ) cos 2 x = 2 2cos1 x + ( 11 ) tan 2 x = x x 2cos1 2cos1 + ( 12 ) iii> công thức tính theo t = tan x/2 : đặt t = tanx/2 khi đó ta có các công thức biểu diễn sau: sin x = 2 1 2 t t + ( 13 ) cos x = 2 2 1 1 t t + − ( 14 ) tan x = 2 1 2 t t − ( 15 ) c. c«ng thøc biÕn ®æi tÝch thµnh tæng vµ ng îc l¹i i> c«ng thøc biÕn ®æi tÝch thµnh tæng cosx.cosy = 2 1 [ cos ( x - y ) + cos ( x + y ) ] ( 16 ) sinx.siny = 2 1 [ cos ( x - y ) - cos ( x + y ) ] ( 17 ) sinx.cosy = 2 1 [ sin( x - y ) + sin ( x + y ) ] ( 18 ) ii> c«ng thøc biÕn ®æi tæng thµnh tÝch : cosx + cosy = 2cos 2 yx + . cos 2 yx − ( 19 ) cosx - cosy = - 2sin 2 yx + . sin 2 yx − ( 20 ) sinx + siny = 2sin 2 yx + . cos 2 yx − ( 21 ) sinx - siny = 2cos 2 yx + . sin 2 yx − ( 22 ) tanx + tany = yx yx cos.cos )sin( + ( 23 ) tanx - tany = yx yx cos.cos )sin( − ( 24 ) chó ý mét sè c«ng thøc sau : sinx + cosx = 2 .sin( x + π /4 ) ( 25) sinx - cosx = 2 .sin( x - π /4 ) ( 26 ) cosx + sinx = 2 .cos( x - π /4 ) ( 27 ) cosx - sinx = 2 .cos( x + π /4 ) ( 28 ) Gi¶i ph ¬ng tr×nh sau : 1. sinx.cosx + | cosx + sinx| = 1 2. 2 2 sinx( x + π /4 ) = 1 1 sin cosx x + 3. 2 + cos2x = - 5sinx 4. 2tanx + cot2x = 2sin2x + 1 sin 2x 5. sin 2 x = cos 2 2x + cos 2 3x 6. 8.cos 3 (x + π /3 ) = cos3x 7. |sinx - cosx| + | sinx + cosx | = 2 8. cos 6 x – sin 6 x = 13/8.cos 2 2x 9. 2sin2x – cos2x = 7.sinx + 2cosx – 4 10. sin3x = cosx.cos2x.( tan 2 x + tan2x ) 11. 4.cos 5 x.sinx – 4sin 5 x.cosx = sin 2 4x 12. sinx.cos4x – sin 2 2x = 4sin 2 ( π /4 – x/2) – 7/2 13. 4cos 3 x + 3 2 .sin2x = 8cosx 14. tanx + 2cot2x =sin2x 15. sin 2 x .sinx - cos 2 x .sin 2 x + 1 = 2.cos 2 ( π /4 - 2 x ) 16. 2.cos 2 x + 2cos 2 2x + 2cos 2 3x – 3 = cos4x(2sin2x + 1) 17. 4(sin 4 x + cos 4 x ) + 3 sin4x = 2 18. 1 + cot2x = 2 1 cos 2 sin 2 x x − 19. sin4x – cos4x = 1 + 4 2 sin( x - π /4 ) 20. ( 1 – tanx )( 1 + sin2x) = 1 + tanx 21. 3(sin tan ) 2cos 2 tan sin x x x x x + − = − 22. sin 2 x + sin 2 3x – 3cos 2 2x = 0 23. 4cos 2 x – cos3x = 6cosx – 2( 1 + cos2x) 24. sin3x + cos2x = 1 + 2sinx.cos2x 25. sin2x + 4( cosx – sinx) = 4 26. 3sinx + 2cosx = 2 + 3tanx 27. cos2x + cos3x/4 – 2 = 0 28. 2sin3x - 1 1 2cos3 sin cos x x x = + 29. 2 3.sin 2 2cos 2 2 2 cos 2x x x= − + 30. 2 2 sin x + 2tan 2 x + 5tanx + 5cotx + 4 = 0 31. tan2x + sin2x = 3/2.cotx 32. sin 3 sin 5 3 5 x x = 33. sin( 3 1 3 ) sin( ) 10 2 2 10 2 x x π π − = + 34. sinx – 4 sin 3 x + cosx = 0 35. sinx.sin2x + sin3x = 6cos 3 x 36. 2cosx.cos2x = 1 + cos2x + cos3x 37. 5( sinx + cos3 sin 3 ) cos 2 3 1 2sin 2 x x x x + = + + 38. sin 2 3x – cos 2 4x = sin 2 5x – cos 2 6x 39. cos3x – 4cos2x + 3cosx – 4 = 0 40. cotx – 1 = 2 cos 2 1 sin sin 2 1 tan 2 x x x x + − + 41. cotx – tanx + 4sinx = 2 sin 2x 42. sin 2 ( 2 2 ) tan cos 0 2 4 2 x x x π − − = 43. 5sinx – 2 = 3( 1 – sinx)tan 2 x 44. ( 2cosx – 1)(2sinx + cosx) = sin2x – sinx 45. cos 2 3x.cos2x – cos 2 x = 0 46. 1 + sinx + cosx + sin2x + cos2x = 0 47. cos 4 x + sin 4 x + cos( x - 4 π ).sin(3x - 4 π ) - 3 2 = 0 48. ( cos2x – cos4x ) 2 = 6 + 2sin3x 49. ( cos2x – cos4x) 2 = 6 + 2sin3x 50. 3 sinx + cosx = 1 cos x 51. ( 1 + cosx ).( 1 + sinx ) = 2 52. 2cosx + 2 sin10x = 3 2 + 2cos28x.sinx 53. sin2x + cos2x = 1 + sinx – 4cosx 54. ( 1 cos cosx x− + ).cos2x = 1 2 sin4x 55. 1 2(cos sin ) tan cot 2 cot 1 x x x x x − = + − 56. 4 4 sin cos 1 (tan sin 2 2 x x x x + = + cotx ) 57. sin2x + 2tanx = 3 58. sin 3 ( x + 4 π ) = 2 sinx 59. 8 2 cos 6 x + 2 2 sin 3 x.sin3x - 6 2 cos 4 x – 1 = 0. 60. 1 – 5sinx + 2cos 2 x = 0. tho¶ m·n cosx ≥ 0. 61. cos 3 x + sin 3 x = sin2x + sinx + cosx 62. sinx.cos4x + 2sin 2 2x = 1 – 4.sin 2 ( 4 π - 2 x ) 63. 4 3 sinx.cosx.cos2x = sin8x 64. sin4x – cos4x = 1 + 4(sinx – cosx ) 65. sin( 3x - 4 π ) = sin2x.sin( x + 4 π ) 66. 4sin 3 x.cos3x + 4cos 3 x.sin3x + 3 3 cos4x = 3. 67. 2 2 4 cos cos 3 0 1 tan x x x − = − 68. sin 2 4x – cos 2 6x = sin( 10,5 π + 10x) 69. tan 2 x.cot 2 x.cot3x = tan 2 x – cot 2 x + cot3x 70. sin3x + 2cos2x – 2 = 0. 71. cos2x + 3cosx + 2 = 0 72. 3cos4x – 2cos 2 3x = 1. 73. 1 + 3cosx + cos2x = cos3x + 2sinx.sin2x 74. tanx + tan2x = - sin3x.cos2x 75. 3( cotx – cosx ) – 5(tanx – sinx) = 2 76. tanx + cotx = 2( sin2x + cos2x ) 77. sin 4 x + cos 4 x = 7 8 cotg( x + 3 π ).cotg( ) 6 x π − 78. 2 2 ( sinx + cosx ).cosx = 3 + cos2x 79.sin 4 x + sin 4 ( x + 4 π ) + sin 4 (x - 4 π ) = 9 8 80. sin 2 2 1 sin x x + + cosx = 0 81. cos 2 x + sinx – 3sin 2 x.cosx = 0 82. 2sin 3 x + cos2x = sinx 83. 3 cos cos 1 2x x− − + = 84. sinx.cosx + 2sinx + 2cosx = 2 85. sin3x(cosx – 2sin3x) + cos3x(1 + sinx – 2cos3x) = 0. 86. 3 5 4sin( ) 2 3 sin x x π + − = − 87. 3tan 3 x – tanx + 2 2 3(1 sin ) 8cos ( ) cos 4 2 x x x π + − − = 0. 88. cos7x - 3 sin7x = - 2 , 2 6 5 7 x π π < < 89.cosx.cos2x.cos4x.cos8x = 1 16 90. 2cos 3 x = sin3x 91. cos2x - 3 sin2x - 3 sinx – cosx + 4 = 0 92. cos2x = cos 2 x. 1 tan x+ 93. 3cot 2 x + 2 2 sin 2 2x = ( 2 + 3 2 )cosx 94.tanx – sin2x – cos2x + 2(2cosx - 1 cos x ) = 0 95. 4( sin3x – cos2x) = 5(sinx – 1) 96.2cos2x + sin 2 x.cosx + sinx.cos 2 x = sinx + cosx 97. tanx.sin 2 x -2sin 2 x = 3( cos2x + sinx.cosx) 98.sin2x( cotx + tan2x) = 4cos 2 x 99. 48 - 4 2 1 2 (1 cot 2 .cot ) 0 cos sin x x x x − + = 100. sin 6 x + cos 6 x = cos4x 101. cos 3 x + cos 2 x + 2sinx – 2 = 0 102. 2 + cosx = 2tan 2 x 103. cos3x + 2 2 2 cos 3 2(1 sin 2 )x x− = + 104. sinx + sin2x + sin3x = 0 105. cotx – tanx = sinx + cosx 106.sin3x + cos2x =1 + 2sinx.cos2x 107. 2cos2x – 8cosx + 7 = 1 cos x 108. cos3x.cos 3 x – sin3x.sin 3 x = cos 3 4x + 1 4 109. 9sinx + 6cosx -3sin2x + cos2x = 8 110. sin 3 x.cos3x + cos 3 x.sin3x = sin 3 4x 111. sin 8 x + cos 8 x = 2( sin 10 x + cos 10 x ) + 5 4 cos2x 112. 2 4 sin 2 cos 2 1 sin .cos x x x x + − = 0 113. 2sin 3 x – cos2x + cosx = 0 114. 1 + cos 3 x – sin 3 x = sin2x 115. 2 sin sin sin cos 1x x x x+ + + = 116. cos 2 x + cos 2 2x + cos 2 3x + cos 2 4x = 3 2 117. cosx + cos2x + cos3x + cos4x = 0 118. 3sinx + 2cosx = 2 + 3tanx. 119. 6 6 2(cos sin ) sin .cos 0 2 2sin x x x x x + − = − 120. cotx + sinx( 1 + tanx.tan 2 x ) = 4. 121. cos3x + cos2x – cosx – 1 = 0 122. (1 + sin 2 x).cosx + (1 + cos 2 x).sinx = 1 + sin2x 123. 2sin 2 2x + sin7x – 1 = sinx 124. ( sin 2 x + cos 2 x ) 2 + 3 .cosx = 3 125. sin2x + cos2x -3sinx – cosx + 1 = 0 126. sin3x - 3 cos3x = 2sin2x 127. sin 3 x - 3 cos 3 x = sinx.cos 2 x - 3 sin 2 x.cosx 128. 2sinx( 1 + cos2x ) + sin2x = 1 + 2cosx 129. 1 1 7 4sin( ) 3 sin 4 sin( ) 2 x x x π π + = − − 130. sin 3 x + cos 3 x + cos2x = 0 131. sin( 5 2 4 x π − ) – cos( 2 4 x π − ) = 3 2 cos 2 x 132. 1 + tan 2 x = cosx + tan² 2 x       − + 1xcos xsin x2cos 133. tanx + 2cot2x = cosx + sin2x 134. tan 4 x + 1 2 4 (2 sin 2 )sin 3 cos x x x − = 135. − = − 1 2(cos sin ) tanx + cot2x cot 1 x x x 136. 9sinx + 6 cosx + cos2x - 3sin2x = 8 137. cos3x.sin2x - cos4x.sinx = xx cos13sin 2 1 ++ 138. 2 1 2 3 sin. 2 sin.sin 2 3 cos. 2 cos.cos =− xx x xx x 139. 2sinx + cosx = sin2x + 1 140. 1 sin 1 x cotx cosx − = − + 141. 2 2 cos x 2 3 sin x cos x 1 3(sin x 3 cos x)+ + = + 142. 1 1 sin 2x sin x 2cot 2x 2sin x sin 2x + − − = 143. ( ) tan 3 cot 8 cos 2 sin 3 cosx x x x x- = + 144. ( ) ( ) 01cos232tan1sin2 222 =−+− xxx 145. cotx = tanx + x x 2sin 4cos2 146. 4 4 cos sin cos sin− = +x x x x 147. cos 3 x + sin 3 x + 1 = 2.cos 2 x 148. Sinx + tanx = Cosx 1 + Cos(x - π ) 149. 3 34 cos.sin2cot.costan.sin 22 =−+ xxxxxx 150. 2 2 (1 cos 2 ) sin 2.cos 2 2sin 2 x x x x + + = 151. 2 1 2sin 3 2 sin sin 2 1 2sin cos 1 x x x x x + − + = − 152. ( ) ( ) 2 2 2 2sin x 1 tan 2x 3 2 cos x 1 0.− + − = 153. 2sin 2x 4sin x 1 0. 6 π   − + + =  ÷   154. cos 2x (1 2cos x)(sin x cos x) 0.+ + − = 155. 3 2 4sin x 4sin x 3sin2x 6cosx 0.+ + + = 156. 3 3 2 cos x sin x 2sin x 1.+ + = 157.sin2x + sinx - 1 1 2cot 2 2sin sin 2 x x x − = 158. 2cos 2 x + 2 3 sinx.cosx + 1 = 3(sinx + 3 cosx) 159.(2sin 2 x -1)tan 2 2x + 3(2cos 2 x – 1 ) = 0 160. sin 2 cos 2 tan cot cos sin x x x x x x + = − . . < < + - - - III < < 3 /2 - - + + IV 3 /2 < < 2 - + - - + > sin = 1 = /2 + k2 ; k Z +> sin = - 1 = - /2 + k2 ;. - ) = sin cos ( - ) = - cos tan( - ) = - tan cot( - ) = - cot iv> cung phụ nhau : sin ( /2 - ) = cos cos ( /2 - ) = sin tan ( /2 -