Question Number 168737 by infinityaction last updated on 16/Apr/22
Commented by infinityaction last updated on 16/Apr/22
$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{x}} \\ $$
Answered by mr W last updated on 17/Apr/22
$${AD}={BC}=\mathrm{1} \\ $$$$\frac{{BD}}{\mathrm{sin}\:\mathrm{54}°}=\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{84}°} \\ $$$$\frac{\mathrm{sin}\:{x}}{{BD}}=\frac{\mathrm{sin}\:\mathrm{30}°}{\mathrm{1}} \\ $$$$\mathrm{sin}\:{x}=\frac{\mathrm{sin}\:\mathrm{54}°\:\mathrm{sin}\:\mathrm{30}°}{\mathrm{sin}\:\mathrm{84}°} \\ $$$$\Rightarrow{x}=\mathrm{24}° \\ $$
Commented by infinityaction last updated on 17/Apr/22
$${sir}\:{solve}\:{this}\:{equation}\:\frac{\mathrm{sin54}°\:\:\mathrm{sin30}°\:}{\mathrm{sin}\:\mathrm{84}°} \\ $$
Commented by som(math1967) last updated on 17/Apr/22
$${sin}\mathrm{24}\frac{{sin}\mathrm{54}{sin}\mathrm{30}}{{sin}\mathrm{24}{sin}\mathrm{84}} \\ $$$${sin}\mathrm{24}\left(\frac{\mathrm{2}{sin}\mathrm{54}{cos}\mathrm{60}}{\mathrm{2}{sin}\mathrm{24}{cos}\mathrm{6}}\right) \\ $$$$={sin}\mathrm{24}\left(\frac{{sin}\mathrm{114}−{sin}\mathrm{6}}{{sin}\mathrm{30}+{sin}\mathrm{18}}\right) \\ $$$$={sin}\mathrm{24}\left(\frac{{sin}\mathrm{66}−{sin}\mathrm{6}}{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{5}}−\mathrm{1}}{\mathrm{4}}}\right) \\ $$$$={sin}\mathrm{24}\left(\frac{\mathrm{2}{sin}\mathrm{30}{cos}\mathrm{36}}{\frac{\sqrt{\mathrm{5}}+\mathrm{1}}{\mathrm{4}}}\right) \\ $$$$={sin}\mathrm{24}×\frac{{cos}\mathrm{36}}{{cos}\mathrm{36}}={sin}\mathrm{24} \\ $$$$\therefore{sinx}={sin}\mathrm{24}\Rightarrow\boldsymbol{{x}}=\mathrm{24} \\ $$