Question

Если cos(ą)=2/5,3п/2<ą<2п,то найти cot(ą/2)​

Answer 1

$Cos\alpha=\frac{2}{5}\\\\\frac{3\pi }{2}<\alpha<2\pi \\\\\frac{3\pi }{4}<\frac{\alpha }{2} <\pi \RightarrowCtg\frac{\alpha }{2}<0\\\\Ctg\frac{\alpha }{2}=-\sqrt{\frac{1+Cos\alpha }{1-Cos\alpha}}=-\sqrt{\frac{1+\frac{2}{5}}{1-\frac{2}{5}}}=-\sqrt{\frac{7}{5}:\frac{3}{5}}=-\sqrt{\frac{7}{5}*\frac{5}{3}}=-\sqrt{\frac{7}{3}}=\\\\=-\frac{\sqrt{7}}{\sqrt{3}}=-\frac{\sqrt{21} }{3}\\\\Otvet:\boxed{Ctg\frac{\alpha}{2}=-\frac{\sqrt{21}}{3}}$