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³ç³»ï¼ sinα/cosαï¼tanαï¼secα/cscα cosα/sinαï¼cotαï¼cscα/secα å¹³æ¹å
³ç³»ï¼ sin^2(α)ï¼cos^2(α)ï¼1 1ï¼tan^2(α)ï¼sec^2(α) 1ï¼cot^2(α)ï¼csc^2(α)
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sin² α+cos² α=1 tan α *cot α=1
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ï¼sina+sinθï¼*ï¼sina+sinθï¼=sinï¼a+θï¼*sinï¼a-Î¸ï¼ è¯æï¼ï¼sina+sinθï¼*ï¼sina+sinθï¼=2 sin[(θ+a)/2] cos[(a-θ)/2] *2 cos[(θ+a)/2] sin[(a-θ)/2] =sinï¼a+θï¼*sinï¼a-θï¼
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æ£å¼¦ï¼ sin α=â αç对边/â α çæè¾¹ ä½å¼¦ï¼cos α=â αçé»è¾¹/â αçæè¾¹ æ£åï¼tan α=â αç对边/â αçé»è¾¹ ä½åï¼cot α=â αçé»è¾¹/â αç对边
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æ£å¼¦ sin2A=2sinA·cosA ä½å¼¦ 1.Cos2a=Cos^2(a)-Sin^2(a) =2Cos^2(a)-1 =1-2Sin^2(a) 2.Cos2a=1-2Sin^2(a) 3.Cos2a=2Cos^2(a)-1 æ£å tan2A=ï¼2tanAï¼/ï¼1-tan^2(A)ï¼
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sin3α=4sinα·sin(Ï/3+α)sin(Ï/3-α) cos3α=4cosα·cos(Ï/3+α)cos(Ï/3-α) tan3a = tan a · tan(Ï/3+a)· tan(Ï/3-a) ä¸åè§å
¬å¼æ¨å¯¼ sin(3a) =sin(a+2a) =sin2acosa+cos2asina =2sina(1-sin²a)+(1-2sin²a)sina =3sina-4sin^3a cos3a =cos(2a+a) =cos2acosa-sin2asina =(2cos²a-1)cosa-2(1-cos^a)cosa =4cos^3a-3cosa sin3a=3sina-4sin^3a =4sina(3/4-sin²a) =4sina[(â3/2)²-sin²a] =4sina(sin²60°-sin²a) =4sina(sin60°+sina)(sin60°-sina) =4sina*2sin[(60+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°-a)/2] =4sinasin(60°+a)sin(60°-a) cos3a=4cos^3a-3cosa =4cosa(cos²a-3/4) =4cosa[cos²a-(â3/2)^2] =4cosa(cos²a-cos²30°) =4cosa(cosa+cos30°)(cosa-cos30°) =4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*{-2sin[(a+30°)/2]sin[(a-30°)/2]} =-4cosasin(a+30°)sin(a-30°) =-4cosasin[90°-(60°-a)]sin[-90°+(60°+a)] =-4cosacos(60°-a)[-cos(60°+a)] =4cosacos(60°-a)cos(60°+a) ä¸è¿°ä¸¤å¼ç¸æ¯å¯å¾ tan3a=tanatan(60°-a)tan(60°+a)
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sinï¼n aï¼=Rsina sinï¼a+Ï/nï¼â¦â¦sinï¼a+ï¼n-1ï¼Ï/nï¼ã å
¶ä¸R=2^ï¼n-1ï¼ è¯æï¼å½sinï¼naï¼=0æ¶ï¼sina=sinï¼Ï/nï¼æ=sinï¼2Ï/nï¼æ=sinï¼3Ï/nï¼æ=â¦â¦æ=sinãï¼n-1ï¼Ï/nã è¿è¯´æsinï¼naï¼=0ä¸ï½sina-sinï¼Ï/nï¼ï½*ï½sina-sinï¼2Ï/nï¼ï½*ï½sina-sinï¼3Ï/nï¼ï½*â¦â¦*ï½sina- sinãï¼n-1ï¼Ï/nã=0æ¯å解æ¹ç¨ã æ以sinï¼naï¼ä¸ï½sina-sinï¼Ï/nï¼ï½*ï½sina-sinï¼2Ï/nï¼ï½*ï½sina-sinï¼3Ï/nï¼ï½*â¦â¦*ï½sina- sinãï¼n-1ï¼Ï/nãææ£æ¯ã èï¼sina+sinθï¼*ï¼sina+sinθï¼=sinï¼a+θï¼*sinï¼a-θï¼ï¼æ以 ï½sina-sinï¼Ï/nï¼ï½*ï½sina-sinï¼2Ï/nï¼ï½*ï½sina-sinï¼3Ï/nï¼ï½*â¦â¦*ï½sina- sinãï¼n-1Ï/nã ä¸sina sinï¼a+Ï/nï¼â¦â¦sinï¼a+ï¼n-1ï¼Ï/nï¼ææ£æ¯ï¼ç³»æ°ä¸næå
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tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA); cot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA. sin^2(a/2)=(1-cos(a))/2 cos^2(a/2)=(1+cos(a))/2 tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))
åå·®å积
sinθ+sinÏ = 2 sin[(θ+Ï)/2] cos[(θ-Ï)/2]
sinθ-sinÏ = 2 cos[(θ+Ï)/2] sin[(θ-Ï)/2] cosθ+cosÏ = 2 cos[(θ+Ï)/2] cos[(θ-Ï)/2] cosθ-cosÏ = -2 sin[(θ+Ï)/2] sin[(θ-Ï)/2] tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB) tanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)
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cos(α+β)=cosαcosβ-sinαsinβcos(α-β)=cosαcosβ+sinαsinβsin(α+β)=sinαcosβ+cosαsinβsin(α-β)=sinαcosβ -cosαsinβ
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sinαsinβ = [cos(α-β)-cos(α+β)] /2 cosαcosβ = [cos(α+β)+cos(α-β)]/2 sinαcosβ = [sin(α+β)+sin(α-β)]/2 cosαsinβ = [sin(α+β)-sin(α-β)]/2
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sinh(a) = [e^a-e^(-a)]/2 cosh(a) = [e^a+e^(-a)]/2 tanh(a) = sin h(a)/cos h(a) å
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³ç³»ï¼ sinï¼Ï/2+αï¼= cosα cosï¼Ï/2+αï¼= -sinα tanï¼Ï/2+αï¼= -cotα cotï¼Ï/2+αï¼= -tanα sinï¼Ï/2-αï¼= cosα cosï¼Ï/2-αï¼= sinα tanï¼Ï/2-αï¼= cotα cotï¼Ï/2-αï¼= tanα sinï¼3Ï/2+αï¼= -cosα cosï¼3Ï/2+αï¼= sinα tanï¼3Ï/2+αï¼= -cotα cotï¼3Ï/2+αï¼= -tanα sinï¼3Ï/2-αï¼= -cosα cosï¼3Ï/2-αï¼= -sinα tanï¼3Ï/2-αï¼= cotα cotï¼3Ï/2-αï¼= tanα (以ä¸kâZ) A·sin(Ït+θ)+ B·sin(Ït+Ï) = â{(A² +B² +2ABcos(θ-Ï)} · sin{ Ït + arcsin[ (A·sinθ+B·sinÏ) / â{A^2 +B^2; +2ABcos(θ-Ï)} } âè¡¨ç¤ºæ ¹å·,å
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sin(-α) = -sinα cos(-α) = cosα tan (-α)=-tanα sin(Ï/2-α) = cosα cos(Ï/2-α) = sinα sin(Ï/2+α) = cosα cos(Ï/2+α) = -sinα sin(Ï-α) = sinα cos(Ï-α) = -cosα sin(Ï+α) = -sinα cos(Ï+α) = -cosα tanA= sinA/cosA tanï¼Ï/2ï¼Î±ï¼ï¼ï¼cotα tanï¼Ï/2ï¼Î±ï¼ï¼cotα tanï¼Ïï¼Î±ï¼ï¼ï¼tanα tanï¼Ïï¼Î±ï¼ï¼tanα 诱导å
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sinα=2tan(α/2)/[1+(tan(α/2))²] cosα=[1-(tan(α/2))²]/[1+(tan(α/2))²] tanα=2tan(α/2)/[1-(tan(α/2))²]
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(1) (sinα)²+(cosα)²=1 (2)1+(tanα)²=(secα)² (3)1+(cotα)²=(cscα)² è¯æä¸é¢ä¸¤å¼,åªéå°ä¸å¼,å·¦å³åé¤(sinα)²ï¼ç¬¬äºä¸ªé¤(cosα)²å³å¯ (4)对äºä»»æéç´è§ä¸è§å½¢,æ»æ tanA+tanB+tanC=tanAtanBtanC è¯: A+B=Ï-C tan(A+B)=tan(Ï-C) (tanA+tanB)/(1-tanAtanB)=(tanÏ-tanC)/(1+tanÏtanC) æ´çå¯å¾ tanA+tanB+tanC=tanAtanBtanC å¾è¯ åæ ·å¯ä»¥å¾è¯,å½x+y+z=nÏ(nâZ)æ¶,该å
³ç³»å¼ä¹æç« ç±tanA+tanB+tanC=tanAtanBtanCå¯å¾åºä»¥ä¸ç»è®º (5)cotAcotB+cotAcotC+cotBcotC=1 (6)cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2) (7)(cosAï¼²+(cosBï¼²+(cosCï¼²=1-2cosAcosBcosC (8)ï¼sinAï¼²+ï¼sinBï¼²+ï¼sinCï¼²=2+2cosAcosBcosC å
¶ä»ééç¹ä¸è§å½æ° csc(a) = 1/sin(a) sec(a) = 1/cos(a)
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sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-cosAsinB cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB tan(A+B) = (tanA+tanB)/(1-tanAtanB) tan(A-B) = (tanA-tanB)/(1+tanAtanB) cot(A+B) = (cotAcotB-1)/(cotB+cotA) cot(A-B) = (cotAcotB+1)/(cotB-cotA)
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