Question

Вычислите: сtg (arccos (-1/3) - π). Большо-о-е мерси!

Answer 1

$\displaystyle ctg(arccos(-1/3)-\pi)=ctg(-(\pi-arccos(-1/3))=-ctg(\pi -arccos(-1/3))=\\\\=-(-ctg(arccos(-1/3))=ctg(arccos(-1/3))$

Далее воспользуемся формулой

$\displaystyle ctg^2x+1=\frac{1}{sin^2x}\\\\ctg^2x=\frac{1}{1-cos^2x}-1=\frac{1-1+cos^2x}{1-cos^2x}=\frac{cos^2x}{1-cos^2x}\\\\ctgx=\sqrt{\frac{cos^2x}{1-cos^2x}}=\frac{cosx}{\sqrt{1-cos^2x}}$

Подставим

$\displaystyle ctg(arccos(-1/3))=\frac{cos(arccos(-1/3)}{\sqrt{1-cos^2(arccos(-1/3)}}=\\\\=\frac{-1/3}{\sqrt{1-1/9}}=\frac{-1/3}{\sqrt{8/9}}=-\frac{1}{\sqrt{8}}$