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Question Number 206082 by hardmath last updated on 06/Apr/24
If   cos𝛂 = sin𝛂 + (1/( (√3)))  Find   sin2𝛂 = ?
$$\mathrm{If}\:\:\:\mathrm{cos}\boldsymbol{\alpha}\:=\:\mathrm{sin}\boldsymbol{\alpha}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$$$\mathrm{Find}\:\:\:\mathrm{sin2}\boldsymbol{\alpha}\:=\:? \\ $$
Answered by MATHEMATICSAM last updated on 29/Apr/24
cosα = sinα + (1/( (√3)))  ⇒ cosα − sinα = (1/( (√3)))  ⇒ cos^2 α + sin^2 α − 2sinαcosα = (1/3)  ⇒ 1 − 2sinαcosα = (1/3)   ⇒ 2sinαcosα = 1 − (1/3) = (2/3)  ⇒ sin2α = (2/3)
$$\mathrm{cos}\alpha\:=\:\mathrm{sin}\alpha\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$$$\Rightarrow\:\mathrm{cos}\alpha\:−\:\mathrm{sin}\alpha\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$$$\Rightarrow\:\mathrm{cos}^{\mathrm{2}} \alpha\:+\:\mathrm{sin}^{\mathrm{2}} \alpha\:−\:\mathrm{2sin}\alpha\mathrm{cos}\alpha\:=\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Rightarrow\:\mathrm{1}\:−\:\mathrm{2sin}\alpha\mathrm{cos}\alpha\:=\:\frac{\mathrm{1}}{\mathrm{3}}\: \\ $$$$\Rightarrow\:\mathrm{2sin}\alpha\mathrm{cos}\alpha\:=\:\mathrm{1}\:−\:\frac{\mathrm{1}}{\mathrm{3}}\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\Rightarrow\:\mathrm{sin2}\alpha\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$
Answered by mr W last updated on 06/Apr/24
cos α−sin α=(1/( (√3)))  cos (α+(π/4))=(1/( (√6)))  cos 2(α+(π/4))=2×((1/( (√6))))^2 −1=−(2/3)  cos (2α+(π/2))=−(2/3)  sin (−2α)=−(2/3)  ⇒sin (2α)=(2/3) ✓
$$\mathrm{cos}\:\alpha−\mathrm{sin}\:\alpha=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$$$\mathrm{cos}\:\left(\alpha+\frac{\pi}{\mathrm{4}}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}} \\ $$$$\mathrm{cos}\:\mathrm{2}\left(\alpha+\frac{\pi}{\mathrm{4}}\right)=\mathrm{2}×\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\right)^{\mathrm{2}} −\mathrm{1}=−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{cos}\:\left(\mathrm{2}\alpha+\frac{\pi}{\mathrm{2}}\right)=−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{sin}\:\left(−\mathrm{2}\alpha\right)=−\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\Rightarrow\mathrm{sin}\:\left(\mathrm{2}\alpha\right)=\frac{\mathrm{2}}{\mathrm{3}}\:\checkmark \\ $$

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