已知sin(45º+α)sin(45º-α)=-1/4,0º<α<90º.(1)求α的值(2)求sin(α+10º)[1-√3tan(α-10º)]的值.
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已知sin(45º+α)sin(45º-α)=-1/4,0º<α<90º.(1)求α的值(2)求sin(α+10º)[1-√3tan(α-10º)]的值.
已知sin(45º+α)sin(45º-α)=-1/4,0º<α<90º.
(1)求α的值
(2)求sin(α+10º)[1-√3tan(α-10º)]的值.
已知sin(45º+α)sin(45º-α)=-1/4,0º<α<90º.(1)求α的值(2)求sin(α+10º)[1-√3tan(α-10º)]的值.
1.sin(45º+α)sin(45º-α)=(sin45ºcosα + cos45ºsinα)(sin45ºcosα - cos45ºsinα)
=(√2/2cosα + √2/2sinα)(√2/2cosα - √2/2sinα)
=(√2/2cosα)^2 - (√2/2sinα)^2
=(1/2)[(cosα)^2 - (sinα)^2] = -1/4
(cosα)^2 - (sinα)^2 = -1/2
[1 - (sinα)^2] - (sinα)^2 = -1/2
-2(sinα)^2 = -3/2
(sinα)^2 = 3/4
sinα= ±√3/2
∵0º<α<90º
∴α=60º
2.原式=sin70º(1 -√3tan50º) = [sin(90°-20°)][1 - √3tan(90°-40°)]
=cos20°(1-√3ctg40°)
=cos20°[1-√3(cos40°/sin40°)]
=cos20°[(sin40° / sin40°) - (√3cos40° / sin40°)
=2cos20°{[(1/2)sin40° - (√3/2)cos40°] / (sin40°)}
=2cos20°[(cos60°sin40° - sin60°cos40°)/ (sin40°)]
=2cos20°[sin(40°-60°) / (sin40°)]
=2cos20°(-sin20° / sin40°)
=-sin40° / sin40°
=-1