Sin 10 * sin 20 * sin 30 * sin 40 * cos10 *cos 20 * cos30 * cos40 =. Created by fromnewyork. algebra-ru

теперь вычислим только числитель Ответ

Sin 10 * sin 20 * sin 30 * sin 40 * cos10 *cos 20 * cos30 * cos40 =...

fromnewyork @fromnewyork

July 2022 1 19 Report

Sin 10 * sin 20 * sin 30 * sin 40 * cos10 *cos 20 * cos30 * cos40 =

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Матов

Verified answer

$sin10=cos80\\
sin20=cos70\\
sin30=cos60\\
sin40=cos50\\\\
(cos80*cos10)*(cos70*cos20)*(cos60*cos30)*(cos50*cos40)=\\\\
 \frac{cos90+cos70}{2}*\frac{cos90+cos50}{2}*\frac{cos90+cos30}{2}*\frac{cos90+cos10}{2}=\\\\
 \frac{cos70*cos50*cos30*cos10}{16}=\\\\
\frac{(cos120+cos20)(cos40+cos20)}{64}=\\\\
$
$\frac{(cos20-0.5)(cos40+cos20)}{64}=$ теперь вычислим только числитель
$(cos20+cos120)(cos40+cos20)=\\
4cos70*cos50*cos30*cos10=\\
cos70*cos50*cos10*2\sqrt{3}=\\
(cos20-0.5)*cos10*\sqrt{3}=\\
\frac{(cos20-0.5)*cos10*\sqrt{3}*2sin10}{2sin10}=\\
\frac{2cos20*sin20*\sqrt{3}-sin20*\sqrt{3}}{4sin10}=\\
\frac{\sqrt{3}(sin40-sin20)}{4sin10}=\frac{\sqrt{3}*2sin10*cos30}{4sin10}=\\ 
 \frac{\sqrt{3}*\frac{\sqrt{3}}{2}}{2}=\frac{3}{4}\\
\\
\frac{\frac{3}{4}}{64}=\frac{3}{256}$
Ответ $\frac{3}{256}$

fromnewyork Спасибо большое. Нашла свою ошибку.

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