Question

Упростите:

$sqrt{1+cos2 alpha/2 } +cos alpha$

$cos pi /5+cos2 pi /5+cos4 pi /5+cos6 pi /5$

Вычислите:

cos36*cos72

sin18*cos54

$cos2 pi /7+cos4 pi /7+cos6 pi /7$

Answer 1

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

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