Made by: Nguyễn Hải Duy - Mr.Alvin/bbk_decon C«ng thøc l−îng gi¸c I. Gi¸ trÞ c¸c hµm sè l−îng gi¸c cña c¸c cung (gãc ) ®Æc biÖt Gãc 00 0 Hslg sin α 0 cos α 1 tan α 0 cot α kx® 300 450 00 900 π π π π 6 1 2 4 2 2 2 2 1 3 3 2 1 2 2 1 3 kx® 1 3 3 0 3 2 3 3 3 0 1200 2π 3 3 2 1 − 2 − 3 − 1350 3π 4 2 2 2 − 2 -1 1800 π 3600 2π 0 0 3 2 3 − 3 − 3 -1 1 0 0 kx® kx® − -1 3 3 1500 5π 6 1 2 Ta nªn sö dông ®−êng trßn l−îng gi¸c ®Ó ghi nhí c¸c gi¸ trÞ ®Æc biÖt II. Quan hÖ l−îng gi¸c cña c¸c cung (gãc) cã liªn quan ®Æc biÖt 1. Cung ®èi nhau:( Cos ®èi) cos(−α ) = cos α sin(−α ) = − sin α (α và -α) tan(−α ) = − tan α cot(−α ) = − cot α 2. Cung bï nhau(Sin bï) cos(π − α ) = − cos α sin(π − α ) = sin α (α, π -α) tan(π − α ) = − tan α cot(π − α ) = − cot α 4. Cung phô nhau: (phô chÐo) 5. Cung h¬n kÐm π tan( − α ) = cotα 2 sin(α + k 2π ) = sin α cos( + α ) = − sin α 2 π π 6. Cung h¬n kÐm k 2π (k ∈ Z) 2 π cos( − α ) = sin α 2 sin( − α ) = cos α 2 π 3. Cung h¬n kÐm π : cos(π + α ) = − cos α sin(π + α ) = − sin α (α, π +α) tan(π + α ) = tan α cot(π + α ) = cot α π (α, π 2 -α) π cot( − α ) = tan α 2 sin( + α ) = cos α 2 π (α, tan( + α ) = − cot α 2 π 2 cos(α + k 2π ) = cos α -α) tan(α + k 2π ) = tan α π cot( + α ) = − tan α 2 cot(α + k 2π ) = cot α III. C«ng thøc l−îng gi¸c 1. C«ng thøc l−îng gi¸c c¬ b¶n sin α tan α = cosα cosα cot α = sinα 2. C«ng thøc céng cos(α + β ) = cos α .cos β − sin α .sin β cos(α − β ) = cos α .cos β + sin α .sin β sin(α + β ) = sin α .cos β + sin β .cos α sin(α − β ) = sin α .cos β − sin β .cos α cos 2α + sin 2 α = 1 1 1 + tan α = cos 2α 2 1 sin 2 α tanα . cotα = 1 1 + cot 2α = tanα +tanβ 1 − tan α .tan β tanα − tanβ tan(α − β ) = 1 + tan α . tan β tan(α +β ) = Made by: Nguyễn Hải Duy - Mr.Alvin/bbk_decon 3. C«ng thøc nh©n ®«i sin 2α = 2sin α .cos α sin α = 2sin cos 2α = cos 2 α − sin 2 α 2 tan α tan 2α = 1 − tan 2 α cot 2 α − 1 cot 2α = 2 cot α α .cos = 2 cos α − 1 2 2 1 ± sin 2α = (sin α ± cosα ) 2 = 1 − 2 sin 2 α cosα = cos 2 2 4. C«ng thøc nh©n ba α 2 − sin 2 sin 3α = 3sin α − 4sin 3 α sin 3 α = 3 sin α − sin 3α 4 1 − cos 4α 2 1 − cos 2α tan 2 α = 1 + cos 2α α 2 .cos α −β 2 2 α +β α −β cos α − cos β = −2 sin .sin 2 2 α +β α −β sin α + sin β = 2 sin .cos 2 2 α +β α −β sin α − sin β = 2 cos .sin 2 2 cos 2 2α = 1 + cos 4α 2 (α ≠ π + k 2π ) 2t 1− t2 2t sin α = ; cos α ; tan α = = 2 2 1+ t 1+ t 1+ t2 7. C«ng thøc biÕn ®æi tÝch thµnh tæng 1 cos α .cos β =  cos(α − β ) + cos(α + β ) 2 1 sin α .sin β =  cos(α − β ) − cos(α + β ) 2 1 sin α .cos β = sin(α − β ) + sin(α + β ) 2 8. C«ng thøc biÕn ®æi tæng thµnh tÝch α +β 2 sin 2 2α = 6.C«ng thøc tÝnh sin α ,cos α ,tgα theo t = tan cos α + cos β = 2 cos α 3cos α + cos 3α 4 3 tan α − tan 3 α 1 − 3 tan 2 α 5. C«ng thøc h¹ bËc: 1 + cos 2α cos 2 α = 2 1 − cos 2α sin 2 α = 2 1 sin α .cos α = sin 2α 2 sin 4α = 2sin 2α .cos 2α cos 3α = cos 3α = 4 cos 3 α − 3cos α tan 3α = α tan α = 0 ⇔ α = kπ sin α = 0 ⇔ α = kπ sin α = 1 ⇔ α = π 2 + k 2π sin α = −1 ⇔ α = − cos α = 0 ⇔ α = π tan α = 1 ⇔ α = π 2 + k 2π + kπ 2 cos α = 1 ⇔ α = k 2π cos α = −1 ⇔ α = π + k 2π sin(α + β ) cos α .cos β sin(α − β ) tan α − tan β = cos α .cos β sin(α + β ) cot α + cot β = sin α .sin β sin( β − α ) cot α − cot β = sin α .sin β a 2 a 1 − cos a = 2 sin 2 2 4 + kπ tan α = −1 ⇔ α = − cot α = 0 ⇔ α = cot α = 1 ⇔ α = π 2 π 4 π 4 + kπ π 4 tan α + tan β = α,β ≠ π 2 + kπ ( k ∈ Z ) α , β ≠ kπ ( k ∈ Z ) π sin α + cos α = 2 cos(α − ) = 2 sin(α + ) 4 4 π π sin α − cos α = − 2 cos(α + ) = 2 sin(α − ) 4 4 π π cos α − sin α = 2 cos(α + ) = 2 sin( − α ) 4 4 Once you choose hope, anything’s possible (Dale Carnegie) If you find yourself in a hole, the first thing to do is stop digging (Will Rogers) + kπ + kπ cot α = −1 ⇔ α = − π 1 + cos a = 2 cos 2 π + kπ  ...Made by: Nguyễn Hải Duy - Mr.Alvin/bbk_decon C«ng thøc nh©n ®«i sin 2α = 2sin α cos α sin α = 2sin cos 2α = cos α − sin