task/30681465
Решить уравнение: 3. sin(π/3 +x)*cos2x - cos(π/3 +x)*sin2x = 0,5
решение * sin(α - β) = sinα*cosβ - cosα*sinβ, [ α= π/3+x , β=2x ] *
sin(π/3 +x)*cos2x - cos(π/3 +x)*sin2x =0,5 ⇔ sin(π/3 +x - 2x) = 0,5 ;
sin( -(x -π/3) ) = 0,5 ⇔ - sin(x -π/3) = 0,5 ⇔ sin(x -π/3) = - 0,5 ⇔
x-π/3 = (-1)ⁿ⁺¹π/6 +πn , n ∈ℤ ⇔ x = (-1)ⁿ⁺¹π/6 +π/3πn , n ∈ℤ .
ответ : x = (-1)ⁿ⁺¹π/6 +π/3πn , n ∈ ℤ .
4. Решить уравнение: tg(π/4 - x) =cosx / ( sinx+cosx )
решение * * tg(α - β) = (tgα - tgβ) /(1+ tgα*tgβ) , [ α= π/4 , β = x] * *
tg(π/4 -x) =cosx / ( sinx+cosx ) ⇔
( tg(π/4)- tgx ) / ( 1+ tg(π/4)*tgx) =cosx / ( sinx+cosx ) , [ tg(π/4) = 1 ] ⇔
(1- tgx ) / ( 1+ tgx) = cosx / ( sinx+cosx) [ tgx = sinx /cosx ] ⇔
(cosx- sinx ) / ( cosx +sinx ) = cosx / ( sinx+cosx) ⇔
(cosx- sinx - cosx ) / ( cosx +sinx ) = 0 ⇔ - sinx / ( cosx +sinx ) =0 ⇔
sinx = 0 * * * очевидно sinx + cosx = 0 ± 1 = ± 1 ≠ 0 * * *
x = πn , n ∈ ℤ ответ : x = πn , n ∈ ℤ