Задача 79834 ...
Условие
б) укажите корни этого уравнения, принадлежащие отрезку [3π/2;3π]
математика 10-11 класс
1024
Решение
★
tg^2 x*(tg x - 1) - 3(tg x - 1) = 0
(tg x - 1)(tg^2 x - 3) = 0
tg x1 = 1
[b]x1 = π/4 + π*k, k ∈ Z[/b]
tg^2 x = 3
tg x2 = -sqrt(3)
[b]x2 = -π/3 + π*n, n ∈ Z[/b]
tg x3 = sqrt(3)
[b]x3 = π/3 + π*n, n ∈ Z[/b]
б) Корни, принадлежащие отрезку [3π/2; 3π]
k = 2, x1 = π/4 + 2π = 9π/4 ∈ [6π/4; 12π/4]
n = 2, x2 = -π/3 + 2π = 5π/3 = 10π/6 ∈ [9π/6; 18π/6]
n = 2, x3 = π/3 + 2π = 7π/3 = 14π/6 ∈ [9π/6; 18π/6]
n = 3, x4 = -π/3 + 3π = 8π/3 = 16π/6 ∈ [9π/6; 18π/6]
Ответ: а) x1 = π/4 + π*k, k ∈ Z, x2 = -π/3 + π*n, x3 = π/3 + π*n, n ∈ Z
б) x1 = 9π/4, x2 = 5π/3, x3 = 7π/3, x4 = 8π/3