Число корней уравнения ctg(2х + 120) = корень из 3 на 3, принадлежащих промежутк...

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Число корней уравнения ctg(2х + 120) = корень из 3 на 3, принадлежащих промежутку (-180;
270], равно

A) 0

B) 5

C) 1

D) 4

Answers & Comments

kalbim

Verified answer

Ctg(2x + 120) = √3/3
ctg(2x + 2π/3) = √3/3
2x + 2π/3 = π/3 + πk, k∈Z
2x = π/3 - 2π/3 + πk = -π/3 + πk, k∈Z
x = -π/6 + πk/2, k∈Z
x∈(-π; 3π/2]
-π < -π/6 + πk/2 ≤ 3π/2
-π + π/6 < πk/2 ≤ 3π/2 + π/6
-5/3 < k ≤ 10/3, k∈Z
k = -1, 0, 1, 2, 3 - значит указанному промежутку принадлежит ПЯТЬ корней.

k = 0, x1 = -π/6 = -30 градусов
k = -1, x2 = -π/6 - π/2 = -2π/3 = -120градусов
k = 1, x = -π/6 + π/2 = π/3 = 60градусов
k = 2, x = -π/6 + π = 5π/6 = 150градусов
k = 3, x = -π/6 + 3π/2 = 8π/6 = 4π/3 = 240градусов

Ответ: В) 5 корней

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