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³ç³»ï¼ã ããsinα/cosα=tanα=secα/cscα ããcosα/sinα=cotα=cscα/secα ããå¹³æ¹å
³ç³»ï¼ ããsin^2(α)+cos^2(α)=1 ãã1+tan^2(α)=sec^2(α) ãã1+cot^2(α)=csc^2(α)
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ããsin^2(α)+cos^2(α)=1 ããtan α *cot α=1
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ããï¼sina+sinθï¼*ï¼sina-sinθï¼=sinï¼a+θï¼*sinï¼a-Î¸ï¼ ããè¯æï¼ï¼sina+sinθï¼*ï¼sina-sinθï¼=2 sin[(θ+a)/2] cos[(a-θ)/2] *2 cos[(θ+a)/2] sin[(a-θ)/2] ãã=sinï¼a+θï¼*sinï¼a-θï¼
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ããæ£å¼¦ï¼ sin α=â αç对边/â α çæè¾¹ ããä½å¼¦ï¼cos α=â αçé»è¾¹/â αçæè¾¹ ããæ£åï¼tan α=â αç对边/â αçé»è¾¹ ããä½åï¼cot α=â αçé»è¾¹/â αç对边
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ããæ£å¼¦ ããsin2A=2sinA·cosA ããä½å¼¦ ãã1.cos2a=cos^2(a)-sin^2(a) ãã2.cos2a=1-2sin^2(a) ãã3.cos2a=2cos^2(a)-1 ããå³cos2a=cos^2(a)-sin^2(a)=2cos^2(a)-1=1-2sin^2(a) ããæ£å ããtan2A=ï¼2tanAï¼/ï¼1-tan^2(A)ï¼
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ããsin3α=4sinα·sin(Ï/3+α)sin(Ï/3-α) ããcos3α=4cosα·cos(Ï/3+α)cos(Ï/3-α) ããtan3a = tan a · tan(Ï/3+a)· tan(Ï/3-a) ããä¸åè§å
¬å¼æ¨å¯¼ã ããsin(3a) ãã=sin(a+2a) ãã=sin2acosa+cos2asina ãã=2sina(1-sina)+(1-2sina)sina ãã=3sina-4sin^3a ããcos3a ãã=cos(2a+a) ãã=cos2acosa-sin2asina ãã=(2cosa-1)cosa-2(1-cos^a)cosa ãã=4cos^3a-3cosa ããsin3a=3sina-4sin^3a ãã=4sina(3/4-sina) ãã=4sina[(â3/2)-sina] ãã=4sina(sin60°-sina) ãã=4sina(sin60°+sina)(sin60°-sina) ãã=4sina*2sin[(60+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°-a)/2] ãã=4sinasin(60°+a)sin(60°-a) ããcos3a=4cos^3a-3cosa ãã=4cosa(cosa-3/4) ãã=4cosa[cosa-(â3/2)^2] ãã=4cosa(cosa-cos30°) ãã=4cosa(cosa+cos30°)(cosa-cos30°) ãã=4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*{-2sin[(a+30°)/2]sin[(a-30°)/2]} ãã=-4cosasin(a+30°)sin(a-30°) ãã=-4cosasin[90°-(60°-a)]sin[-90°+(60°+a)] ãã=-4cosacos(60°-a)[-cos(60°+a)] ãã=4cosacos(60°-a)cos(60°+a) ããä¸è¿°ä¸¤å¼ç¸æ¯å¯å¾ ããtan3a=tanatan(60°-a)tan(60°+a) ããç°ååºå
¬å¼å¦ä¸ï¼ã ããsin2α=2sinαcosα tan2α=2tanα/(1-tanα ï¼ cos2α=cosα-sinα=2cosα-1=1-2sinαã ããå¯å«è½»è§è¿äºå符ï¼å®ä»¬å¨æ°å¦å¦ä¹ ä¸ä¼èµ·å°éè¦ä½ç¨ï¼å
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ããsin3α=3sinα-4sinα=4sinα·sin(Ï/3+α)sin(Ï/3-α) cos3α=4cosα-3cosα=4cosα·cos(Ï/3+α)cos(Ï/3-α) tan3α=tan(α)*(-3+tan(α)^2)/(-1+3*tan(α)^2)=tan a · tan(Ï/3+a)· tan(Ï/3-a)
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ããsin^2(α/2)=(1-cosα)/2 cos^2(α/2)=(1+cosα)/2 tan^2(α/2)=(1-cosα)/(1+cosα) tan(α/2)=sinα/(1+cosα)=(1-cosα)/sinα
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ããsinα=2tan(α/2)/[1+tan(α/2)] cosα=[1-tan(α/2)]/[1+tan^2(α/2)] tanα=2tan(α/2)/[1-tan&s(α/2)]
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ããsinα+sin(α+2Ï/n)+sin(α+2Ï*2/n)+sin(α+2Ï*3/n)+â¦â¦+sin[α+2Ï*(n-1)/n]=0 cosα+cos(α+2Ï/n)+cos(α+2Ï*2/n)+cos(α+2Ï*3/n)+â¦â¦+cos[α+2Ï*(n-1)/n]=0 以å sin^2(α)+sin^2(α-2Ï/3)+sin^2(α+2Ï/3)=3/2 tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0
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ããsin4A=-4*(cosA*sinA*(2*sinA^2-1)) cos4A=1+(-8*cosA^2+8*cosA^4) tan4A=(4*tanA-4*tanA^3)/(1-6*tanA^2+tanA^4)
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ããsin5A=16sinA^5-20sinA^3+5sinA cos5A=16cosA^5-20cosA^3+5cosA tan5A=tanA*(5-10*tanA^2+tanA^4)/(1-10*tanA^2+5*tanA^4)
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ããsin6A=2*(cosA*sinA*(2*sinA+1)*(2*sinA-1)*(-3+4*sinA^2)) cos6A=((-1+2*cosA)*(16*cosA^4-16*cosA^2+1)) tan6A=(-6*tanA+20*tanA^3-6*tanA^5)/(-1+15*tanA-15*tanA^4+tanA^6)
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ããsin7A=-(sinA*(56*sinA^2-112*sinA^4-7+64*sinA^6)) cos7A=(cosA*(56*cosA^2-112*cosA^4+64*cosA^6-7)) tan7A=tanA*(-7+35*tanA^2-21*tanA^4+tanA^6)/(-1+21*tanA^2-35*tanA^4+7*tanA^6)
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ããsin8A=-8*(cosA*sinA*(2*sinA^2-1)*(-8*sinA^2+8*sinA^4+1)) cos8A=1+(160*cosA^4-256*cosA^6+128*cosA^8-32*cosA^2) tan8A=-8*tanA*(-1+7*tanA^2-7*tanA^4+tanA^6)/(1-28*tanA^2+70*tanA^4-28*tanA^6+tanA^8)
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ããsin9A=(sinA*(-3+4*sinA^2)*(64*sinA^6-96*sinA^4+36*sinA^2-3)) cos9A=(cosA*(-3+4*cosA^2)*(64*cosA^6-96*cosA^4+36*cosA^2-3)) tan9A=tanA*(9-84*tanA^2+126*tanA^4-36*tanA^6+tanA^8)/(1-36*tanA^2+126*tanA^4-84*tanA^6+9*tanA^8)
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ããsin10A=2*(cosA*sinA*(4*sinA^2+2*sinA-1)*(4*sinA^2-2*sinA-1)*(-20*sinA^2+5+16*sinA^4)) cos10A=((-1+2*cosA^2)*(256*cosA^8-512*cosA^6+304*cosA^4-48*cosA^2+1)) tan10A=-2*tanA*(5-60*tanA^2+126*tanA^4-60*tanA^6+5*tanA^8)/(-1+45*tanA^2-210*tanA^4+210*tanA^6-45*tanA^8+tanA^10)
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ããæ ¹æ®æ££ç¾å¼å®çï¼(cosθ+ i sinθ)^n = cos(nθ)+ i sin(nθ) 为æ¹ä¾¿æè¿°ï¼ä»¤sinθ=sï¼cosθ=c èèn为æ£æ´æ°çæ
å½¢ï¼ cos(nθ)+ i sin(nθ) = (c+ i s)^n = C(n,0)*c^n + C(n,2)*c^(n-2)*(i s)^2 + C(n,4)*c^(n-4)*(i s)^4 + ... +C(n,1)*c^(n-1)*(i s)^1 + C(n,3)*c^(n-3)*(i s)^3 + C(n,5)*c^(n-5)*(i s)^5 + ... =>æ¯è¾ä¸¤è¾¹çå®é¨ä¸èé¨ å®é¨ï¼cos(nθ)=C(n,0)*c^n + C(n,2)*c^(n-2)*(i s)^2 + C(n,4)*c^(n-4)*(i s)^4 + ... i*(èé¨)ï¼i*sin(nθ)=C(n,1)*c^(n-1)*(i s)^1 + C(n,3)*c^(n-3)*(i s)^3 + C(n,5)*c^(n-5)*(i s)^5 + ... 对ææçèªç¶æ°nï¼ 1. cos(nθ)ï¼ å
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¨é¨é½å¯ä»¥æ¹æ以c(ä¹å°±æ¯cosθ)表示ã 2. sin(nθ)ï¼ (1)å½næ¯å¥æ°æ¶ï¼ å
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¨é¨é½å¯ä»¥æ¹æ以s(ä¹å°±æ¯sinθ)表示ã (2)å½næ¯å¶æ°æ¶ï¼ å
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ããtan(A/2)=(1-cosA)/sinA=sinA/(1+cosA) ããcot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA ããsin^2(a/2)=(1-cos(a))/2 ããcos^2(a/2)=(1+cos(a))/2 ããtan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a)) åè§å
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ããcos(α+β)=cosαcosβ-sinαsinβ ããcos(α-β)=cosαcosβ+sinαsinβ ããsin(α+β)=sinαcosβ+cosαsinβ ããsin(α-β)=sinαcosβ -cosαsinβ ããtan(α+β)=(tanα+tanβ)/(1-tanαtanβ) ããtan(α-β)=(tanα-tanβ)/(1+tanαtanβ) ããcot(A+B) = (cotAcotB-1)/(cotB+cotA) ããcot(A-B) = (cotAcotB+1)/(cotB-cotA)
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ããsin(α+β+γ)=sinα·cosβ·cosγ+cosα·sinβ·cosγ+cosα·cosβ·sinγ-sinα·sinβ·sinγ ããcos(α+β+γ)=cosα·cosβ·cosγ-cosα·sinβ·sinγ-sinα·cosβ·sinγ-sinα·sinβ·cosγ ããtan(α+β+γ)=(tanα+tanβ+tanγ-tanα·tanβ·tanγ)/(1-tanα·tanβ-tanβ·tanγ-tanγ·tanα)
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ããsinθ+sinÏ = 2 sin[(θ+Ï)/2] cos[(θ-Ï)/2] åå·®å积å
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sinθ-sinÏ = 2 cos[(θ+Ï)/2] sin[(θ-Ï)/2] ããcosθ+cosÏ = 2 cos[(θ+Ï)/2] cos[(θ-Ï)/2] ããcosθ-cosÏ = -2 sin[(θ+Ï)/2] sin[(θ-Ï)/2] ããtanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB) ããtanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)
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ããsinαsinβ =-[cos(α+β)-cos(α-β)] /2 ããcosαcosβ = [cos(α+β)+cos(α-β)]/2 ããsinαcosβ = [sin(α+β)+sin(α-β)]/2 ããcosαsinβ = [sin(α+β)-sin(α-β)]/2
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ããsh a = [e^a-e^(-a)]/2 ããch a = [e^a+e^(-a)]/2 ããth a = sin h(a)/cos h(a) ããå
¬å¼ä¸ï¼ ãã设α为任æè§ï¼ç»è¾¹ç¸åçè§çåä¸ä¸è§å½æ°çå¼ç¸çï¼ ããsinï¼2kÏ+αï¼= sinα ããcosï¼2kÏ+αï¼= cosα ããtanï¼2kÏ+αï¼= tanα ããcotï¼2kÏ+αï¼= cotα ããå
¬å¼äºï¼ ãã设α为任æè§ï¼Ï+αçä¸è§å½æ°å¼ä¸Î±çä¸è§å½æ°å¼ä¹é´çå
³ç³»ï¼ ããsinï¼Ï+αï¼= -sinα ããcosï¼Ï+αï¼= -cosα ããtanï¼Ï+αï¼= tanα ããcotï¼Ï+αï¼= cotα ããå
¬å¼ä¸ï¼ ããä»»æè§Î±ä¸ -αçä¸è§å½æ°å¼ä¹é´çå
³ç³»ï¼ ããsinï¼-αï¼= -sinα ããcosï¼-αï¼= cosα ããtanï¼-αï¼= -tanα ããcotï¼-αï¼= -cotα ããå
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³ç³»ï¼ ããsinï¼Ï-αï¼= sinα ããcosï¼Ï-αï¼= -cosα ããtanï¼Ï-αï¼= -tanα ããcotï¼Ï-αï¼= -cotα ããå
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¬å¼ä¸å¯ä»¥å¾å°2Ï-αä¸Î±çä¸è§å½æ°å¼ä¹é´çå
³ç³»ï¼ ããsinï¼2Ï-αï¼= -sinα ããcosï¼2Ï-αï¼= cosα ããtanï¼2Ï-αï¼= -tanα ããcotï¼2Ï-αï¼= -cotα ããå
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ï¼ ããÏ/2±αå3Ï/2±αä¸Î±çä¸è§å½æ°å¼ä¹é´çå
³ç³»ï¼ ããsinï¼Ï/2+αï¼= cosα ããcosï¼Ï/2+αï¼= -sinα ããtanï¼Ï/2+αï¼= -cotα ããcotï¼Ï/2+αï¼= -tanα ããsinï¼Ï/2-αï¼= cosα ããcosï¼Ï/2-αï¼= sinα ããtanï¼Ï/2-αï¼= cotα ããcotï¼Ï/2-αï¼= tanα ããsinï¼3Ï/2+αï¼= -cosα ããcosï¼3Ï/2+αï¼= sinα ããtanï¼3Ï/2+αï¼= -cotα ããcotï¼3Ï/2+αï¼= -tanα ããsinï¼3Ï/2-αï¼= -cosα ããcosï¼3Ï/2-αï¼= -sinα ããtanï¼3Ï/2-αï¼= cotα ããcotï¼3Ï/2-αï¼= tanα ãã(以ä¸kâZ) ããA·sin(Ït+θ)+ B·sin(Ït+Ï) = ããâ{(A+2ABcos(θ-Ï)} · sin{Ït + arcsin[ (A·sinθ+B·sinÏ) / â{A^2 +B^2 +2ABcos(θ-Ï)} } ããâè¡¨ç¤ºæ ¹å·,å
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¬å¼ä¸ï¼ã ããsin(-α) = -sinα ããcos(-α) = cosα ããtan (-α)=-tanα ããå
¬å¼äºï¼ ããsin(Ï/2-α) = cosα ããcos(Ï/2-α) = sinα ããå
¬å¼ä¸ï¼ ããsin(Ï/2+α) = cosα ããcos(Ï/2+α) = -sinα ããå
¬å¼åï¼ ããsin(Ï-α) = sinα ããcos(Ï-α) = -cosα ããå
¬å¼äºï¼ ããsin(Ï+α) = -sinα ããcos(Ï+α) = -cosα ããå
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ï¼ ããtanA= sinA/cosA ããtanï¼Ï/2+αï¼=ï¼cotα ããtanï¼Ï/2ï¼Î±ï¼=cotα ããtanï¼Ïï¼Î±ï¼=ï¼tanα ããtanï¼Ï+αï¼=tanα ãã诱导å
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ããsinα=2tan(α/2)/[1+(tan(α/2))] ããcosα=[1-(tan(α/2))]/[1+(tan(α/2)] ããtanα=2tan(α/2)/[1-(tan(α/2))]
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ãã(1) (sinα)^2+(cosα)^2=1ï¼å¹³æ¹åå
¬å¼ï¼ ãã(2)1+(tanα)^2=(secα)^2 ãã(3)1+(cotα)^2=(cscα)^2 ããè¯æä¸é¢ä¸¤å¼ï¼åªéå°ä¸å¼,å·¦å³åé¤(sinα)^2ï¼ç¬¬äºä¸ªé¤(cosα)^2å³å¯ ãã(4)对äºä»»æéç´è§ä¸è§å½¢ï¼æ»æ ããtanA+tanB+tanC=tanAtanBtanC ããè¯ï¼ ããA+B=Ï-C ããtan(A+B)=tan(Ï-C) ãã(tanA+tanB)/(1-tanAtanB)=(tanÏ-tanC)/(1+tanÏtanC) ããæ´çå¯å¾ ããtanA+tanB+tanC=tanAtanBtanC ããå¾è¯ ããåæ ·å¯ä»¥å¾è¯,å½x+y+z=nÏ(nâZ)æ¶ï¼è¯¥å
³ç³»å¼ä¹æç« ããç±tanA+tanB+tanC=tanAtanBtanCå¯å¾åºä»¥ä¸ç»è®º ãã(5)cotAcotB+cotAcotC+cotBcotC=1 ãã(6)cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2) ãã(7)(cosA)^2+(cosB)^2+(cosC)^2=1-2cosAcosBcosC ãã(8)ï¼sinA)^2+(sinB)^2+(sinC)^2=2+2cosAcosBcosC ããå
¶ä»ééç¹ä¸è§å½æ°ã ããcsc(a) = 1/sin(a) ããsec(a) = 1/cos(a) ãã(seca)^2+(csca)^2=(seca)^2(csca)^2 ããå¹çº§æ°å±å¼å¼ ããsin x = x-x^3/3!+x^5/5!-â¦â¦+(-1)^(k-1)*(x^(2k-1))/(2k-1)!+â¦â¦ã (-â<x<â) ããcos x = 1-x^2/2!+x^4/4!-â¦â¦+(-1)k*(x^(2k))/(2k)!+â¦â¦ (-â<x<â) ããarcsin x = x + 1/2*x^3/3 + 1*3/(2*4)*x^5/5 + â¦â¦(|x|<1) ããarccos x = Ï - ( x + 1/2*x^3/3 + 1*3/(2*4)*x^5/5 + â¦â¦ ) (|x|<1) ããarctan x = x - x^3/3 + x^5/5 -â¦â¦(xâ¤1) ããæ éå
¬å¼ ããsinx=x(1-x^2/Ï^2)(1-x^2/4Ï^2)(1-x^2/9Ï^2)â¦â¦ ããcosx=(1-4x^2/Ï^2)(1-4x^2/9Ï^2)(1-4x^2/25Ï^2)â¦â¦ ããtanx=8x[1/(Ï^2-4x^2)+1/(9Ï^2-4x^2)+1/(25Ï^2-4x^2)+â¦â¦] ããsecx=4Ï[1/(Ï^2-4x^2)-1/(9Ï^2-4x^2)+1/(25Ï^2-4x^2)-+â¦â¦] ãã(sinx)x=cosx/2cosx/4cosx/8â¦â¦ ãã(1/4)tanÏ/4+(1/8)tanÏ/8+(1/16)tanÏ/16+â¦â¦=1/Ï ããarctan x = x - x^3/3 + x^5/5 -â¦â¦(xâ¤1) ããåèªåéæ°åæ±åæå
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¬å¼ ããsinx+sin2x+sin3x+â¦â¦+sinnx=[sin(nx/2)sin((n+1)x/2)]/sin(x/2) ããcosx+cos2x+cos3x+â¦â¦+cosnx=[cos((n+1)x/2)sin(nx/2)]/sin(x/2) ããtan((n+1)x/2)=(sinx+sin2x+sin3x+â¦â¦+sinnx)/(cosx+cos2x+cos3x+â¦â¦+cosnx) ããsinx+sin3x+sin5x+â¦â¦+sin(2n-1)x=(sinnx)^2/sinx ããcosx+cos3x+cos5x+â¦â¦+cos(2n-1)x=sin(2nx)/(2sinx)
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