已知α，β都是锐角，sinα＝45，cos(α+β)＝513，求sinβ的值．

Question

sinα＝(4)/(5)，cos(α+β)＝(5)/(13)，求sinβ的值．(4)/(5)4455cos(α+β)＝(5)/(13)，求sinβ的值．(5)/(13)551313

Answer 1

∵0＜α＜(π)/(2)，0＜β＜(π)/(2)，sinα＝(4)/(5)，cos(α+β)＝(5)/(13)
∴0＜α+β＜πcosα＝√(1−sin2α)＝√(1−(16)/(25))＝(3)/(5)sin(α+β)＝√(1−cos2(α+β))＝√(1−(25)/(169))＝(12)/(13)
∴sinβ=sin[（α+β）-α]=sin（α+β）cosα-cos（α+β）sinα=(12)/(13)×(3)/(5)−(5)/(13)×(4)/(5)＝(16)/(65) 0＜α＜(π)/(2)πππ222，0＜β＜(π)/(2)πππ222，sinα＝(4)/(5)，cos(α+β)＝(5)/(13)
∴0＜α+β＜πcosα＝√(1−sin2α)＝√(1−(16)/(25))＝(3)/(5)sin(α+β)＝√(1−cos2(α+β))＝√(1−(25)/(169))＝(12)/(13)
∴sinβ=sin[（α+β）-α]=sin（α+β）cosα-cos（α+β）sinα=(12)/(13)×(3)/(5)−(5)/(13)×(4)/(5)＝(16)/(65) sinα＝(4)/(5)444555，cos(α+β)＝(5)/(13)555131313
∴0＜α+β＜πcosα＝√(1−sin2α)＝√(1−(16)/(25))＝(3)/(5)sin(α+β)＝√(1−cos2(α+β))＝√(1−(25)/(169))＝(12)/(13)
∴sinβ=sin[（α+β）-α]=sin（α+β）cosα-cos（α+β）sinα=(12)/(13)×(3)/(5)−(5)/(13)×(4)/(5)＝(16)/(65) cosα＝√(1−sin2α)

1−sin2α

1−sin2α1−sin2α2α＝√(1−(16)/(25))

1−(16)/(25)

1−(16)/(25)1−(16)/(25)161616252525＝(3)/(5)333555sin(α+β)＝√(1−cos2(α+β))＝√(1−(25)/(169))＝(12)/(13)
∴sinβ=sin[（α+β）-α]=sin（α+β）cosα-cos（α+β）sinα=(12)/(13)×(3)/(5)−(5)/(13)×(4)/(5)＝(16)/(65) sin(α+β)＝√(1−cos2(α+β))

1−cos2(α+β)

1−cos2(α+β)1−cos2(α+β)2(α+β)＝√(1−(25)/(169))

1−(25)/(169)

1−(25)/(169)1−(25)/(169)252525169169169＝(12)/(13)121212131313
∴sinβ=sin[（α+β）-α]=sin（α+β）cosα-cos（α+β）sinα=(12)/(13)×(3)/(5)−(5)/(13)×(4)/(5)＝(16)/(65) (12)/(13)121212131313×(3)/(5)333555−(5)/(13)555131313×(4)/(5)444555＝(16)/(65)161616656565