Question

已知α，β都是锐角，sinα=4/5，cos（α+β）=5/13，求sinβ的值

Answer 1

已知α，β都是锐角
sinα=4/5,cos（α+β）=5/13
则cosα=√[1-(4/5)^2]=3/5
sin(α+β)=√[1-(5/13)^2]=12/13

那么sinβ=sin[(α+β)-α]=sin(α+β)cosα-cos(α+β)sinα=(12/13)*(3/5)-(5/13)*(4/5)=16/65

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Answer 2

sina=4/5
cosa=3/5
a和b是锐角
cos（a+b）=5/13说明a+b还是锐角
sin（a+b）=12/13
sinb=sin(a+b-a)=sin(a+b)cosa-cos(a+b)sina=12/13×3/5-5/13×4/5=36/65-20/65=16/65

Answer 3

cos（α+β）=5/13
cosαcosβ+ sinαsinβ = 5/13
(3/5)cosβ + (4/5)sinβ = 5/13
39cosβ + 52sinβ = 25
(39cosβ)^2 = (25-52sinβ)^2
2743(sinβ)^2-2600sinβ + 586=0
sinβ = (2600+ √330408) / 5486