作业帮已知α,β为锐角,且3sin2α+2sin2β=1,3sin2α-2sin2β=0,试求cos(π/3+α+2β)的值.
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已知α,β为锐角,且3sin2α+2sin2β=1,3sin2α-2sin2β=0,试求cos(π/3+α+2β)的值.
已知α,β为锐角,且3sin2α+2sin2β=1,3sin2α-2sin2β=0,试求cos(π/3+α+2β)的值.
已知α,β为锐角,且3sin2α+2sin2β=1,3sin2α-2sin2β=0,试求cos(π/3+α+2β)的值.
α,β为锐角
0 < sinα < 1, 0 < cosα < 1.
0 < sinβ < 1, 0 < cosβ < 1.
3sin2α+2sin2β=1,
3sin2α-2sin2β=0
sin2α = 1/6,
sin2β = 1/4.
(sinα + cosα)^2 = 1 + sin2α = 1 + 1/6 = 7/6,
sinα + cosα = (7/6)^(1/2),
(sinα - cosα)^2 = 1 - sin2α = 1 - 1/6 = 5/6,
sinα - cosα = (5/6)^(1/2),
sinα = [7^(1/2) + 5^(1/2)]/[2(6)^(1/2)]
cosα = [7^(1/2) - 5^(1/2)]/[2(6)^(1/2)].
同理,
sinβ = [5^(1/2) + 3^(1/2)]/4
cos2β = 1 - 2(sinβ)^2 = 1 - 2[5^(1/2) + 3^(1/2)]^2/16
= 1 - [8 + 2(15)^(1/2)]/8
= -(15)^(1/2)/4
sin(π/3 + 2β) = sin(π/3)cos(2β) + cos(π/3)sin(2β)
= 3^(1/2)/2[-(15)^(1/2)/4] + (1/2)(1/4)
= [1 - 3(5)^(1/2)]/8
cos(π/3 + 2β) = cos(π/3)cos(2β) - sin(π/3)sin(2β)
= (1/2)[-(15)^(1/2)/4] - [3^(1/2)/2](1/4)
= -3^(1/2)[1 + 5^(1/2)]/8
cos(π/3 + α + 2β) = cos(π/3 + 2β)cosα - sin(π/3 + 2β)sinα
= {-3^(1/2)[1 + 5^(1/2)]/8}[7^(1/2) - 5^(1/2)]/[2(6)^(1/2)] - {[1 - 3(5)^(1/2)]/8}[7^(1/2) + 5^(1/2)]/[2(6)^(1/2)]
= {-21^(1/2) + 15^(1/2) - 105^(1/2) + 5(3)^(1/2) - 7^(1/2) + 5^(1/2) + 3(35)^(1/2) + 15}/[16(6)^(1/2)]