arcsin(sin 33П/7)+arccos(cos 46П/7)=? В пять действий.. Created by AlinkaMalinkaKalinka. algebra-ru

7)=? В пять действий...

AlinkaMalinkaKalinka @AlinkaMalinkaKalinka

July 2022 1 6 Report

arcsin(sin 33П/7)+arccos(cos 46П/7)=?
В пять действий.

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NNNLLL54

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$arcsin(sinx)=x$ , если $-\dfrac{\pi}{2}\leq x\leq \dfrac{\pi}{2}$

$arccosx(cosx)=x$ , если $0\leq x\leq \pi$

Поэтому надо преобразовать углы к таким, которые лежат в указанных промежутках.

$\displaystyle sin\frac{33\pi}{7}=sin(4\pi +\frac{5\pi }{7})=sin\frac{5\pi}{7}=(\frac{5\pi}{7}\ \textgreater \ \frac{\pi}{2})=sin(\pi -\frac{2\pi}{7})=\\\\=sin\frac{2\pi}{7}\; ,\; \; \; \; -\frac{\pi}{2} \leq \frac{2\pi }{7} \leq \frac{\pi }{2} \\\\\\cos\frac{46\pi}{7}=cos(6\pi + \frac{4\pi}{7}) =cos\frac{4\pi}{7}\; ,\; \; \; 0 \leq \frac{4\pi}{7} \leq \pi$

$\displaystyle arcsin(sin\frac{33\pi}{7})+arccos (cos\frac{46\pi}{7})=arcsin(sin \frac{2\pi }{7} } )+arccos(cos \frac{4\pi}{7} )=\\\\=\frac{2\pi}{7}+ \frac{4\pi}{7}=\frac{6\pi }{7}$

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