
已知π/2<α<π,0<β<π/2,tanα=-3/4,cos(β-α)=5/13,则sinβ的值
已知π/2<α<π,0<β<π/2,tanα=-3/4,cos(β-α)=5/13,则sinβ的值是多少...
已知π/2<α<π,0<β<π/2,tanα=-3/4,cos(β-α)=5/13,则sinβ的值是多少
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∵(secα)^2=1+(tanα)^2=1+(-3/4)^2=25/16
∴secα=±5/4
∵cosα=1/secα
∴cosα=±4/5
∵π/2<α<π
∴cosα=-4/5
则sinα=√[1-(cosα)^2]=3/5
sin(β-α)=±√1-[cos(β-α)]^2=±12/13
∵π/2<α<π,0<β<π/2
∴β-α<0
∴sin(β-α)=-12/13
sinβ=sin[(β-α)+α]=sin(β-α)cosα + cos(β-α)sinα =63/65
∴secα=±5/4
∵cosα=1/secα
∴cosα=±4/5
∵π/2<α<π
∴cosα=-4/5
则sinα=√[1-(cosα)^2]=3/5
sin(β-α)=±√1-[cos(β-α)]^2=±12/13
∵π/2<α<π,0<β<π/2
∴β-α<0
∴sin(β-α)=-12/13
sinβ=sin[(β-α)+α]=sin(β-α)cosα + cos(β-α)sinα =63/65
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