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July 2022 1 15 Report

Упростите:

[tex] \sqrt{1+cos2 \alpha/2 } +cos \alpha [/tex]

[tex]cos \pi /5+cos2 \pi /5+cos4 \pi /5+cos6 \pi /5[/tex]

Вычислите:

cos36*cos72

sin18*cos54

[tex] cos2 \pi /7+cos4 \pi /7+cos6 \pi /7[/tex]

Answers & Comments

Матов

Verified answer

$1)\sqrt{\frac{1+cos2a}{2}}+cosa=\sqrt{\frac{1+2cos^2a-1}{2}}+cosa=|cosa|+cosa=2cosa\\
cosa>0\\
2)cos36+cos72+cos144+cos216\\
1)cos36+cos216=2cos(144)*cos90=0\\
2)cos72+cos144=2cos(108)*cos(36)=2cos(180-72)*cos36=-2cos72*cos36=\frac{-2cos72*cos36*sin36}{sin36}=\frac{-cos72*sin72}{sin36}=\frac{-sin144}{2sin36}=\frac{-sin36}{2sin36}=-\frac{1}{2}$
$3)cos36*cos72=\frac{cos36*cos72*2sin36}{2sin36}=\frac{sin72*cos72}{2sin36}=\frac{sin144}{4sin36}=\frac{sin36}{4sin36}=\frac{1}{4}\\
4)sin18*cos54=sin18*cos(90-36)=sin18*sin36=\frac{2cos18*sin18*sin36}{2cos18}=\frac{sin36*sin36}{2cos18}$

$cos(\frac{2\pi}{7})+cos(\frac{4\pi}{7})+cos(\frac{6\pi}{7})\\
\frac{sin(\frac{\pi}{7})cos(\frac{2\pi}{7})+sin(\frac{\pi}{7})cos(\frac{4\pi}{7})+sin(\frac{\pi}{7})cos(\frac{6\pi}{7})}{sin(\frac{\pi}{7}})=-\frac{1}{2}$

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