Question Number 149292 by mathdanisur last updated on 04/Aug/21
$${if}\:\:\:{sin}\mathrm{2}\boldsymbol{\alpha}\:=\:-\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$${find}\:\:\:\mathrm{3}{tg}^{\mathrm{2}} \left(\frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{4}}\:-\:\boldsymbol{\alpha}\right)\:=\:? \\ $$
Answered by Ar Brandon last updated on 04/Aug/21
$$\mathrm{sin2}\alpha=−\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\Theta=\mathrm{3tan}^{\mathrm{2}} \left(\frac{\mathrm{3}\pi}{\mathrm{4}}−\alpha\right) \\ $$$$\:\:\:\:=\mathrm{3}\left(\frac{\mathrm{tan}\alpha+\mathrm{1}}{\mathrm{tan}\alpha−\mathrm{1}}\right)^{\mathrm{2}} =\mathrm{3}\left(\frac{\mathrm{sin}\alpha+\mathrm{cos}\alpha}{\mathrm{sin}\alpha−\mathrm{cos}\alpha}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:=\mathrm{3}\frac{\mathrm{1}+\mathrm{sin2}\alpha}{\mathrm{1}−\mathrm{sin2}\alpha}=\mathrm{3}\left(\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{3}}{\mathrm{4}}\right)=\frac{\mathrm{3}}{\mathrm{2}} \\ $$
Commented by mathdanisur last updated on 04/Aug/21
$${Thank}\:{You}\:{Ser} \\ $$