Question

вычислить без таблиц:
cos п/11 + cos 3п/11 + cos 5п/11 + cos 7п/11 + cos 9п/11

Answer 1

$cosfrac{pi}{11}+cosfrac{3pi}{11}+cosfrac{5pi}{11}+cosfrac{7pi}{11}+cosfrac{9pi}{11}=\\ frac{2sinfrac{pi}{11}(cosfrac{pi}{11}+cosfrac{3pi}{11}+cosfrac{5pi}{11}+cosfrac{7pi}{11}+cosfrac{9pi}{11})}{2sinfrac{pi}{11}}=\\ frac{sinfrac{2pi}{11}+2sinfrac{pi}{11}*cosfrac{3pi}{11}+2sinfrac{pi}{11}cosfrac{5pi}{11}+\\ 2sinfrac{pi}{11}*cosfrac{7pi}{11}+2sinfrac{pi}{11}*cosfrac{9pi}{11}}{2sinfrac{pi}{11}}=$

$2sinfrac{pi}{11}*cosfrac{3pi}{11}=-sinfrac{2pi}{11}+sinfrac{4pi}{11}\ 2sinfrac{pi}{11}*cosfrac{5pi}{11}=-sinfrac{4pi}{11}+sinfrac{6pi}{11}\ 2sinfrac{pi}{11}*cosfrac{7pi}{11}=-sinfrac{6pi}{11}+sinfrac{8pi}{11}\ 2sinfrac{pi}{11}*cosfrac{9pi}{11}=-sinfrac{8pi}{11}+sinfrac{10pi}{11}\\ frac{sinfrac{10pi}{11}}{2sinfrac{2pi}{11}}=frac{1}{2}$
 Ответ $frac{1}{2}$