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推导公式
tanα cotα=2/sin2α
tanα-cotα=-2cot2α
1 cos2α=2cos^2α
1-cos2α=2sin^2α
1 sinα=(sinα/2 cosα/2)^2
=2sina(1-sin2a) (1-2sin2a)sina
=3sina-4sin3a
cos3a
=cos(2a a)
=cos2acosa-sin2asina
=(2cos2a-1)cosa-2(1-sin2a)cosa
=4cos3a-3cosa
sin3a=3sina-4sin3a
=4sina(3/4-sin2a)
=4sina[(√3/2)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=4cos3a-3cosa
=4cosa(cos2a-3/4)
=4cosa[cos2a-(√3/2)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)
上述两式相比可得
tan3a=tanatan(60°-a)tan(60° a)
半角公式
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))
三角和
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)