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已知cos(π/6+a)cos(π/3-a)=1/4,a属于(π/3,π/2).求.sina和tana-1/tana分别为多少.
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已 知cos(π/6+a)cos(π/3-a)=1/4,a属于(π/3,π/2).求.sina和tana-1/tana分别为多少.
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答案和解析
2cos(π/6+a)cos(π/3-a)=-1/2,
积化和差得cos(π/2)+cos(2a-π/6)=-1/2,
∴cos(2a-π/6)=-1/2,
a∈(π/3,π/2),u=2a-π/6的值域是(π/2,5π/6),
∴2a-π/6=2π/3,a=5π/12.
∴sin2a=1/2,
tana-1/tana
=sina/cosa-cosa/sina
=[(sina)^2-(cosa)^2]/[sinacosa]
=-2cos2a/sin2a
=-2/tan2a
=-2/(-1/√3)
=2√3.
积化和差得cos(π/2)+cos(2a-π/6)=-1/2,
∴cos(2a-π/6)=-1/2,
a∈(π/3,π/2),u=2a-π/6的值域是(π/2,5π/6),
∴2a-π/6=2π/3,a=5π/12.
∴sin2a=1/2,
tana-1/tana
=sina/cosa-cosa/sina
=[(sina)^2-(cosa)^2]/[sinacosa]
=-2cos2a/sin2a
=-2/tan2a
=-2/(-1/√3)
=2√3.
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