18. tg²(3t+π/2)=1/3

Ответ: t = -π/6 + πn/3, n ∈ Z или t = -π/3 + πn/3, n ∈ Z

• tg²(3t + π/2) = 1/3
• tg(3t + π/2) = ±√(1/3)
• tg(3t + π/2) = ±√3/3

Решаем уравнение tg(3t + π/2) = √3/3:

• 3t + π/2 = arctg(√3/3) + πn, n ∈ Z
• 3t + π/2 = π/6 + πn, n ∈ Z
• 3t = π/6 - π/2 + πn, n ∈ Z
• 3t = -π/3 + πn, n ∈ Z
• t = -π/9 + πn/3, n ∈ Z

Решаем уравнение tg(3t + π/2) = -√3/3:

• 3t + π/2 = arctg(-√3/3) + πn, n ∈ Z
• 3t + π/2 = -π/6 + πn, n ∈ Z
• 3t = -π/6 - π/2 + πn, n ∈ Z
• 3t = -2π/3 + πn, n ∈ Z
• t = -2π/9 + πn/3, n ∈ Z

Ответ: t = -π/9 + πn/3, n ∈ Z или t = -2π/9 + πn/3, n ∈ Z