Question

(sin 11П/18 - sin П/18)/(cos 11П/18 - cos П/18)

Answer 1

Формулы, которые нам понадобятся:

$boxed{diplaystyle sin alpha-sin beta=2sin frac{alpha-beta}{2} cdot cos frac{alpha+beta}{2}}$
$boxed{displaystyle cos alpha - cos beta =-2sin frac{alpha+beta}{2} cdot sin frac{alpha-beta}{2}}$


$displaystyle frac{sin frac{11 pi}{18}-sin frac{pi}{18}}{cos frac{11 pi}{18}-cos frac{pi}{18}}=frac{2sin frac{frac{11 pi}{18}- frac{pi}{18}}{2} cdot cos frac{frac{11 pi}{18}+frac{pi}{18}}{2}}{-2sin frac{frac{11 pi}{18}+ frac{pi}{18}}{2} cdot sin frac{frac{11 pi}{18}- frac{pi}{18}}{2}}=$
$displaystyle = frac{2sin (frac{10 pi}{18} cdot frac{1}{2}) cdot cos (frac{12 pi}{18} cdot frac{1}{2})}{-2sin( frac{12 pi}{18} cdot frac{1}{2}) cdot sin(frac{10 pi}{18}cdot frac{1}{2})}=frac{2sin frac{5 pi}{18}cdot cos frac{pi}{3}}{-2sin frac{pi}{3} cdot sin frac{5 pi}{18}}=$
$displaystyle = frac{cos frac{pi}{3}}{-sin frac{pi}{3}}=-ctg frac{pi}{3}=boxed{- frac{sqrt 3}{3}}$