三角函数的公式及其诱导公式
两角和与差的三角函数 cos(α+β)=cosα・cosβ-sinα・sinβ cos(α-β)=cosα・cosβ+sinα・sinβ sin(α±β)=sinα・cosβ±cosα・sinβ tan(α+β)=(tanα+tanβ)/(1-tanα・tanβ) tan(α-β)=(tanα-tanβ)/(1+tanα・tanβ)和差化积公式 sinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2] sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2] cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2] cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2]积化和差公式 sinα・cosβ=(1/2)[sin(α+β)+sin(α-β)] cosα・sinβ=(1/2)[sin(α+β)-sin(α-β)] cosα・cosβ=(1/2)[cos(α+β)+cos(α-β)] sinα・sinβ=-(1/2)[cos(α+β)-cos(α-β)]倍角公式 sin(2α)=2sinα・cosα=2/(tanα+cotα) cos(2α)=(cosα)^2-(sinα)^2=2(cosα)^2-1=1-2(sinα)^2 tan(2α)=2tanα/(1-tan^2α) cot(2α)=(cot^2α-1)/(2cotα) sec(2α)=sec^2α/(1-tan^2α) csc(2α)=1/2*secα・cscα三倍角公式 sin(3α) = 3sinα-4sin^3α = 4sinα・sin(60°+α)sin(60°-α) cos(3α) = 4cos^3α-3cosα = 4cosα・cos(60°+α)cos(60°-α) tan(3α) = (3tanα-tan^3α)/(1-3tan^2α) =tanαtan(π/3+α)tan(π/3-α) cot(3α)=(cot^3α-3cotα)/(3cot^2α-1)n倍角公式 sin(nα)=ncos^(n-1)α・sinα-C(n,3)cos^(n-3)α・sin^3α+C(n,5)cos^(n-5)α・sin^5α-… cos(nα)=cos^nα-C(n,2)cos^(n-2)α・sin^2α+C(n,4)cos^(n-4)α・sin^4α-…半角公式 sin(α/2)=±√((1-cosα)/2) cos(α/2)=±√((1+cosα)/2) tan(α/2)=±√((1-cosα)/(1+cosα))=sinα/(1+cosα)=(1-cosα)/sinα cot(α/2)=±√((1+cosα)/(1-cosα))=(1+cosα)/sinα=sinα/(1-cosα) sec(α/2)=±√((2secα/(secα+1)) csc(α/2)=±√((2secα/(secα-1))辅助角公式 Asinα+Bcosα=√(A^2+B^2)sin(α+φ)(tanφ=B/A) Asinα+Bcosα=√(A^2+B^2)cos(α-φ)(tanφ=A/B)万能公式 sin(a)= (2tan(a/2))/(1+tan^2(a/2)) cos(a)= (1-tan^2(a/2))/(1+tan^2(a/2)) tan(a)= (2tan(a/2))/(1-tan^2(a/2))降幂公式 sin^2α=(1-cos(2α))/2=versin(2α)/2 cos^2α=(1+cos(2α))/2=covers(2α)/2 tan^2α=(1-cos(2α))/(1+cos(2α))三角和的三角函数 sin(α+β+γ)=sinα・cosβ・cosγ+cosα・sinβ・cosγ+cosα・cosβ・sinγ-sinα・sinβ・sinγ cos(α+β+γ)=cosα・cosβ・cosγ-cosα・sinβ・sinγ-sinα・cosβ・sinγ-sinα・sinβ・cosγ tan(α+β+γ)=(tanα+tanβ+tanγ-tanα・tanβ・tanγ)/(1-tanα・tanβ-tanβ・tanγ-tanγ・tanα)其它公式 1+sin(a)=(sin(a/2)+cos(a/2))^21-sin(a)=(sin(a/2)-cos(a/2))^2 csc(a)=1/sin(a) sec(a)=1/cos(a) cos30=sin60 sin30=cos60推导公式 tanα+cotα=2/sin2α tanα-cotα=-2cot2α 1+cos2α=2cos^2α 1-cos2α=2sin^2α 1+sinα=[sin(α/2)+cos(α/2)]^2其他及证明 sinα+sin(α+2π/n)+sin(α+2π*2/n)+sin(α+2π*3/n)+……+sin[α+2π*(n-1)/n]=0 cosα+cos(α+2π/n)+cos(α+2π*2/n)+cos(α+2π*3/n)+……+cos[α+2π*(n-1)/n]=0 以及 sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2 tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0 cosx+cos2x+...+cosnx= [sin(n+1)x+sinnx-sinx]/2sinx 证明: 左边=2sinx(cosx+cos2x+...+cosnx)/2sinx =[sin2x-0+sin3x-sinx+sin4x-sin2x+...+sinnx-sin(n-2)x+sin(n+1)x-sin(n-1)x]/2sinx (积化和差) =[sin(n+1)x+sinnx-sinx]/2sinx=右边 等式得证 sinx+sin2x+...+sinnx= - [cos(n+1)x+cosnx-cosx-1]/2sinx 证明: 左边=-2sinx[sinx+sin2x+...+sinnx]/(-2sinx) =[cos2x-cos0+cos3x-cosx+...+cosnx-cos(n-2)x+cos(n+1)x-cos(n-1)x]/(-2sinx) =- [cos(n+1)x+cosnx-cosx-1]/2sinx=右边 等式得证 三倍角公式推导 sin3a =sin(2a+a) =sin2acosa+cos2asina =2sina(1-sin^2a)+(1-2sin^2a)sina =3sina-4sin^3a cos3a =cos(2a+a) =cos2acosa-sin2asina =(2cos^2a-1)cosa-2(1-cos^2a)cosa =4cos^3a-3cosa sin3a=3sina-4sin^3a =4sina(3/4-sin^2a) =4sina[(√3/2)^2-sin^2a] =4sina(sin^260°-sin^2a) =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(cos^2a-3/4) =4cosa[cos^2a-(√3/2)^2] =4cosa(cos^2a-cos^230°) =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) 上述两式相比可得 tan3a=tanatan(60°-a)tan(60°+a)