已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),且0＜α＜π,0＜β＜π,求α,β的值.

已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),且0＜α＜π,0＜β＜π,求α,β的值.
已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),
已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),且0＜α＜π,0＜β＜π,求α,β的值.

已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),已知cos(π/2-α)=√2cos(3π/2+β),√3sin(3π/2-α)=-√2sin(π/2+β),且0＜α＜π,0＜β＜π,求α,β的值.
由cos(π/2-α)=√2cos(3π/2+β）,得sinα=√2sinβ,(1)
√3sin(3π/2-α)=-√2sin(π/2+β),得√3cosα=√2cosβ,(2)
根据cosα的平方+sinα的平方=1且cosβ的平方+sinβ的平方=1
得cosα的平方*2=1,cosα==正负√2/2 α=π/4或3π/4
由 (2)得 当α=π/4 时,β=π/3
当α=3π/4 时,β=2π/3

因为cos(π/2-α)=√2cos(3π/2+β),所以cosα=√2cos(2π-π/2+β)=√2cos(-π/2+β)=√2cos(π/2-β)=√2cosβ。即cosα=√2cosβ (1)
因为√3sin(3π/2-α)=-√2sin(π/2+β),所以√3sin(2π-π/2-α)=-√2sin(π/2+β),即√3sin(-π/2-α)=-√2sinβ,即-√3sin...

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因为cos(π/2-α)=√2cos(3π/2+β),所以cosα=√2cos(2π-π/2+β)=√2cos(-π/2+β)=√2cos(π/2-β)=√2cosβ。即cosα=√2cosβ (1)
因为√3sin(3π/2-α)=-√2sin(π/2+β),所以√3sin(2π-π/2-α)=-√2sin(π/2+β),即√3sin(-π/2-α)=-√2sinβ,即-√3sin(π/2+α)=-√2sinβ,即-√3sinα=-√2sinβ，即√3sinα=√2sinβ，即sinα=√6/3sinβ (2)
根据cosα的平方+sinα的平方=1,由(1)(2)解得cosβ=正负1/2,所以β=π/3或者2π/3
由(1)知道,当β=π/3时,α=π/4
当β=2π/3时,α=3π/4

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