Решите уравнение tg ((п/2) + x) - tg (2 п-x) = (2√3) / 3

Решение

tg ((п/2) + x) - tg (2 п-x) = (2√3) / 3

- ctgx + tgx = (2√3) / 3

1/tgx - tgx = (2√3) / 3

3tg²x + 2√3tgx - 3 = 0

tgx = t

3t² + 2√3t - 3 = 0

D = 12 + 4*3*3 = 48

t₁ = (- 2√3 - 4√3) / 6

t₁ = - 6√3 / 6

t₁ = - √3

t₂ = (- 2√3 + 4 √3) / 6

t₂ = 2√3 / 6

t₂ = √3/3

1) tgx = - √3

x = arctg (- √3) + πk, k ∈ Z

x₁ = - π/3 + πk, k ∈ Z

2) tgx = √3/3

x = arctg (√3/3) + πn, n ∈ Z

x₂ = π/6 + πn, n ∈ Z

Ответ: x₁ = - π/3 + πk, k ∈ Z; x₂ = π/6 + πn, n ∈ Z