Question

Cos5*cos55*cos65=? Полное решение, пожалуйста! Дано всё в градусах (а не в радианах)

Answer 1

Cos5°*cos55°*cos65° 
* *  cosα*cosβ =(1/2)(cos(α+β) +cos(α -β))_преобраз.  произведен.  в сумму * *
* * * cos(180° -α) = - cosα _формула приведения * * *
* * * cosα/2 = ± √(1+cosα)/2  _формула половинного угла * * *
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cos5°cos55°cos65°  =(1/2)(cos60° +cos50°)cos65° =(1/2)(1/2+cos50°)cos65°
=(1/4)cos65° +(1/2)cos50°cos65° =(1/4)cos65° +(1/4)(cos115°+cos15°) =
(1/4)(cos65° +cos115°+cos15°) =(1/4)(cos65° +cos(180°- 65°) +cos15°) =
(1/4)(cos65° -cos 65° +cos15°) =(1/4)cos15°  =(1/4)√((1+cos30°)/2)  =
(1/4)√((1+√3/2)/2)=(1/4)√((2+√3)/4) =(1/8)√(2+√3)  =

(1/8√2)√(4+2√3) =(2+√3)/8√2 =√2(2+√3)/16.

Answer 2

Сos5*1/2(cos(55-65)+cos(55+65))=cos5*1/2(cos10+cos120)=
=1/2*cos5(cos10-cos60)=1/2*cos5*(cos10-1/2)=1/2*cos5cos10-1/4*cos5=
=1/2*1/2(cos(5-10)+cos(5+10))-1/4cos5=1/4*(cos5+cos15)-1/4cos5=
=1/4*cos5+1/2*cos15-1/4*cos5=1/4*cos15