高一有关三角函数的所有公式?

1个回答

  • ·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

    ·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))^2 1-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

    1+sin(a)=(sin(a/2)+cos(a/2))^2 1-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