Question Number 33186 by NECx last updated on 12/Apr/18
$${Find}\:{the}\:{exact}\:{value}\:{of}\:{sin}\theta\:{if} \\ $$$${cos}\theta=\frac{\mathrm{1}}{\mathrm{57}}\:{and}\:\theta\:{is}\:{obtuse} \\ $$
Answered by MJS last updated on 12/Apr/18
$$\mathrm{cos}^{\mathrm{2}} \theta+\mathrm{sin}^{\mathrm{2}} \theta=\mathrm{1} \\ $$$$\mathrm{sin}\:\theta=\pm\sqrt{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \theta}=\pm\sqrt{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{57}^{\mathrm{2}} }}= \\ $$$$=\pm\sqrt{\frac{\mathrm{57}^{\mathrm{2}} −\mathrm{1}}{\mathrm{57}^{\mathrm{2}} }}=\pm\frac{\mathrm{4}\sqrt{\mathrm{203}}}{\mathrm{57}}\:\Rightarrow \\ $$$$\Rightarrow\:\theta\approx\mathrm{89}°\:\vee\:\theta\approx\mathrm{271}° \\ $$$$\mathrm{since}\:\mathrm{both}\:\mathrm{are}\:\mathrm{not}\:\mathrm{obtuse}\:\mathrm{I}\:\mathrm{guess} \\ $$$$\mathrm{it}\:\mathrm{should}\:\mathrm{be}\:\mathrm{cos}\:\theta=−\frac{\mathrm{1}}{\mathrm{57}}\:\mathrm{because} \\ $$$$\mathrm{then}\:\theta\approx\mathrm{91}°\:\vee\:\theta\approx\mathrm{269}° \\ $$$$\mathrm{and}\:\mathrm{sin}\:\theta=\frac{\mathrm{4}\sqrt{\mathrm{203}}}{\mathrm{57}} \\ $$