Task/28344547
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Решить тригонометрические уравнения
1.
cos(3π/4 - 4x) +0,5 = 0 ;
cos(π - (π/4+4x) )= -1/2 ; * * * cos(π -φ) = - cosφ * * *
- cos(4x+π/4) = -1/2 ;
cos(4x+π/4) = 1/2 ;
4x+π/4 = ± π/3 +2πn , n∈ℤ ;
x = - π/16 ± π/12+(π/2)*n , n∈ℤ .
а)
x = - π/16 - π/12+(π/2)*n , n∈ℤ ;
x = - 7π/48 +(π/2)*n , n∈ℤ .
б)
x = - π/16 + π/12+(π/2)*n , n∈ ℤ ;
x = π/48 +(π/2)*n , n∈ℤ.
ответ: x₁ = - 7π/48 +(π/2)*n , n∈ℤ ; x₂ = π/48 +(π/2)*n , n∈ℤ .
* * * * * * * * * * * * * * * * * * * * *
2.
3cos²x + 10cosx +3 =0 квадратное уравнение относительно cosx
D₁=(10/2)² - 3*3 =4²
cosx₁ = (-5 -4)/3 = -3 < -1 → посторонний корень
cosx₂ = (-5+4)/3 = -1/3 .
x = ± arccos( -1/3) +2πn, n∈ℤ * * * x =± (π - arccos( 1/3) ) +2πn , n∈ℤ* *
* * * * * * * * * * * * * * * * * * * *
3.
sin2x= √2sinx ;
2sinxcosx - √2sinx =0 ;
2sinx(cosx -√2 /2 ) = 0 ;
а)
sinx =0 ;
x =πn , n∈ ℤ .
б)
cosx = √2 /2 ;
x = π/4 +2πn , n∈ ℤ .
ответ: x₁ =πn , n∈ ℤ , x₂ = π/4 +2πn , n∈ ℤ .