А) 3 tg^3 x - 2 tg x - 1=0
tgx = t
3t³ - 2t - 1 = 0
t₁ = 1
3t³ - 2t - 1 I t - 1
-(3t³ - 3t²) 3t² + 3t + 1
3t² - 2t
-(3t² - 3t)
t - 1
-(t - 1)
0
3t³ - 2t - 1 = (t - 1)*(3t² + 3t + 1)
3t² + 3t + 1 = 0
D = 9 - 4*3*1 = - 3 < 0
решений нет
tgx = 1
x = π/4 + πk, k∈Z
b) ctg^3 2x - 6 ctg 2x + 5= 0
ctg2x = y
y³ - 6y + 5 = 0
y = 1
y³ - 6y + 5 I y - 1
-(y³ - y²) y² + y - 5
y² - 6y
-(y² - y)
- 5y + 5
- (-5y + 5)
0
y³ - 6y + 5 = (y - 1)*(y² + y - 5)
ctg2x = 1
2x = π/4 + πn, n∈Z
x₁ = π/8 + πn/2, n∈Z
y² + y - 5 = 0
D = 1 + 4*1*5 = 21
y₁ = (- 1 - √21)/2
y₂ = (- 1 + √21)/2
ctg2x = (- 1 - √21)/2
2x = arctg (- 1 - √21)/2) + πm, m∈Z
x₂ = 1/2 arctg (- 1 - √21)/2) + πm/2, m∈Z
ctg2x = (- 1 + √21)/2
2x = arctg (- 1 + √21)/2) + πm, m∈Z
x₃ = 1/2 arctg (- 1 + √21)/2) + πm/2, m∈Z