1) a) cos780°=cos(2·360°+60°)=cos60=1/2=0.5
б) sin(13π/6)=sin(π/2+π/6)=cos(π/6)=(√3)/2
2) найти sin(α) , если cosα=-12/13 , α∈(π;3π/2)
sin(α)= - √(1-cos²α)= - √(1-(-12/13)²)= - √(169-144)/13
3)
a) cos(α-β) - cos(α+β)=[cosα·cosβ+sinαsinβ]-[cosα·cosβ-sinαsinβ]=2sinαsinβ
б) sin(-α)+cos(π+α) -sin(α)-cos(α) -(sinα+cosα)
---------------------------- = ---------------------------------------- =----------------- =
1+2cos(π/2-α)cos(-α) cos²(α)+sin²(α)+2sin(α)cos(α) (sinα+cosα)²
-1/(sinα+cosα)
4) доказать
cos4α+1=(1/2)sin4α·(ctgα-tgα)
(1/2)sinα·(ctgα-tgα)=(1/2)sin4α·(cosα/sinα-sinα/(cosα)=
=(1/2)sin4α·(cos²α-sin²α)/(cosα·sinα)=sin4α·cos2α/(sin2α)=
=2sin2α·cos²2α/(sin2α)=2cos²2α=1+cos4α
cos4α+1=1+cos4α Ч.Т.Д.
5) РЕШИТЬ УРАВНЕНИЕ
sin5xcos4x-sin4xcos5x=1
sin(5x-4x)=1
sin(x)=1 ⇔x=π/2+2πn, n∈Z