483 - 2 484 - 2 и 4 .. Created by Akula17. geometriya-ru

483 - 2 484 - 2 и 4...

Akula17 @Akula17

October 2021 1 26 Report

483 - 2
484 - 2 и 4 .

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Artem112

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483
Так как угол принадлежит 2 четверти, то его синус будет положительным
$\sin \alpha = \sqrt{1-\cos^2 \alpha } = \sqrt{1-(- \frac{1}{3})^2} = \sqrt{ \frac{8}{9}} = \frac{2 \sqrt{2} }{3}$
$\cos( \alpha - \frac{ \pi }{4} )=\cos \alpha \cos \frac{ \pi }{4}+\sin \alpha \sin \frac{ \pi }{4}=
- \frac{1}{3}\cdot \frac{ \sqrt{2} }{2} + \frac{2 \sqrt{2} }{3}\cdot \frac{ \sqrt{2} }{2} =
\\\
=- \frac{ \sqrt{2} }{6} + \frac{4}{6}= \frac{4- \sqrt{2} }{6}$

484
$\cos5 \beta \cos2 \beta +\sin5 \beta \sin2 \beta =\cos(5 \beta -2 \beta )=\cos3 \beta
\\\
\cos( \frac{7 \pi }{5}+ \alpha )\cos( \frac{2 \pi }{5}+ \alpha )+\sin( \frac{7 \pi }{5}+ \alpha )\sin( \frac{2 \pi }{5}+ \alpha )=
\\\
=\cos( \frac{7 \pi }{5}+ \alpha - \frac{2 \pi }{5}- \alpha )=
\cos \pi =-1$

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