Question Number 160746 by Rokon last updated on 05/Dec/21
Answered by som(math1967) last updated on 06/Dec/21
$${if}\:\boldsymbol{\theta}=\mathrm{45}ยฐ\boldsymbol{{then}}\:\boldsymbol{{A}}=\sqrt{\mathrm{2}}\:\boldsymbol{{B}}=\mathrm{0} \\ $$$$\left.{ii}\right)\:{A}=\sqrt{\mathrm{2}}\left({cos}\theta+\boldsymbol{{sin}\theta}โ{sin}\boldsymbol{\theta}\right) \\ $$$$\therefore\boldsymbol{{cos}\theta}+\boldsymbol{{sin}\theta}=\sqrt{\mathrm{2}}\boldsymbol{{cos}\theta} \\ $$$$\Rightarrow\boldsymbol{{sin}\theta}=\left(\sqrt{\mathrm{2}}โ\mathrm{1}\right)\boldsymbol{{cos}\theta} \\ $$$$\Rightarrow\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)\boldsymbol{{sin}\theta}=\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)\left(\sqrt{\mathrm{2}}โ\mathrm{1}\right)\boldsymbol{{cos}\theta} \\ $$$$\Rightarrow\left(\sqrt{\mathrm{2}}+\mathrm{1}\right)\boldsymbol{{sin}\theta}=\boldsymbol{{cos}\theta} \\ $$$$\Rightarrow\sqrt{\mathrm{2}}\boldsymbol{{sin}\theta}=\boldsymbol{{cos}\theta}โ\boldsymbol{{sin}\theta} \\ $$$$\Rightarrow\sqrt{\mathrm{2}}\left({c}\boldsymbol{{os}\theta}+\boldsymbol{{sin}\theta}โ\boldsymbol{{cos}\theta}\right)=\boldsymbol{{B}} \\ $$$$\therefore\boldsymbol{{B}}=\sqrt{\mathrm{2}}\left(\boldsymbol{{A}}โ\boldsymbol{{cos}\theta}\right) \\ $$