Question Number 106220 by bemath last updated on 03/Aug/20
$$\mathrm{find}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{cos}\:\left(\mathrm{x}−\mathrm{45}°\right)=\mathrm{sin}\:\left(\mathrm{2x}+\mathrm{60}°\right) \\ $$
Answered by Dwaipayan Shikari last updated on 03/Aug/20
$${sin}\left(\frac{\pi}{\mathrm{2}}−{x}+\frac{\pi}{\mathrm{4}}\right)={sin}\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{3}}\right) \\ $$$$\frac{\mathrm{3}\pi}{\mathrm{4}}−{x}=\mathrm{2}{k}\pi\pm\mathrm{2}{x}+\frac{\pi}{\mathrm{3}}\:\:\:\:\left({k}\in\mathbb{Z}\right) \\ $$$${first}\:{case} \\ $$$$\mathrm{3}{x}+\mathrm{2}{k}\pi=\frac{\mathrm{5}\pi}{\mathrm{12}} \\ $$$${second}\:{case} \\ $$$${x}=\mathrm{2}{k}\pi+\frac{\mathrm{13}\pi}{\mathrm{12}} \\ $$
Answered by john santu last updated on 03/Aug/20
Answered by malwaan last updated on 03/Aug/20
$${cos}\left({x}−\frac{\pi}{\mathrm{4}}\right)={sin}\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{3}}\right) \\ $$$${sin}\left(\frac{\pi}{\mathrm{2}}−{x}+\frac{\pi}{\mathrm{4}}\right)={sin}\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{3}}\right) \\ $$$${sin}\left(\frac{\mathrm{3}\pi}{\mathrm{4}}−{x}\right)={sin}\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{3}}\right) \\ $$$$\frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{4}}−\boldsymbol{{x}}=\mathrm{2}\boldsymbol{{x}}+\frac{\boldsymbol{\pi}}{\mathrm{3}}\:+\mathrm{2}\boldsymbol{{k}\pi} \\ $$$$\Rightarrow−\mathrm{3}\boldsymbol{{x}}=\frac{−\mathrm{6}\boldsymbol{\pi}}{\mathrm{12}}+\mathrm{2}\boldsymbol{{k}\pi} \\ $$$$\Rightarrow\boldsymbol{{x}}=\:\frac{\boldsymbol{\pi}}{\mathrm{6}}\:−\frac{\mathrm{2}\boldsymbol{\pi{k}}}{\mathrm{3}} \\ $$$$\boldsymbol{{or}} \\ $$$$\frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{4}}−\boldsymbol{{x}}=\boldsymbol{\pi}−\left(\mathrm{2}\boldsymbol{{x}}+\frac{\boldsymbol{\pi}}{\mathrm{3}}\right)+\mathrm{2}\boldsymbol{{k}\pi} \\ $$$$\Rightarrow\boldsymbol{{x}}=\:\mathrm{2}\boldsymbol{{k}\pi}−\frac{\boldsymbol{\pi}}{\mathrm{12}} \\ $$