1 cos2x=1-2sin²x 3-6sin²x-5sinx+1=0 sinx=a 6a²+5a-4=0 D=25+96=121 a1=(-5-11)/12=-4/3⇒sinx=-4/3<-1 нет решения a2=(-5+11)/12=1/2⇒sinx=1/2 x=π/6+2πn U x=5π/6+2πk π≤π/6+2πn≤5π/2 6≤1+12n≤15 5≤12n≤14 5/12≤n≤14/12 n=1⇒x=π/6+2π=13π/6 π≤5π/6+2πk≤5π/2 6≤5+12k≤15 1≤12k≤10 1/12≤k≤10/12 нет решения
log₁₅₀(5) = 1 / log₅(150)
log₅(30) : (1 / log₅(150)) = log₅(30) * log₅(150) = log₅(5*6) * log₅(6*25) =
= ( log₅(5)+log₅(6) ) * ( log₅(25)+log₅(6) ) = ( 1+log₅(6) ) * ( 2+log₅(6) )
аналогично:
log₅(750) : (1 / log₅(6)) = log₅(750) * log₅(6) = log₅(6*125) * log₅(6) =
= ( log₅(125)+log₅(6) ) * log₅(6) = ( 3+log₅(6) ) * log₅(6)
осталось вычесть... удобно обозначить x = log₅(6)
( 1+x ) * ( 2+x ) - ( 3+x ) * x = 2+x+2x+x² - 3x-x² = 2
log₂(70) : (1 / log₂(280)) = log₂(70) * log₂(280) = log₂(35*2) * log₂(35*8) =
= ( log₂(2)+log₂(35) ) * ( log₂(8)+log₂(35) ) = ( 1+log₂(35) ) * ( 3+log₂(35) )
аналогично:
log₂(560) : (1 / log₂(35)) = log₂(560) * log₂(35) = log₂(16*35) * log₂(35) =
= ( log₂(16)+log₂(35) ) * log₂(35) = ( 4+log₂(35) ) * log₂(35)
осталось вычесть... удобно обозначить x = log₂(35)
( 1+x ) * ( 3+x ) - ( 4+x ) * x = 3+x+3x+x² - 4x-x² = 3