18. tg²(3t + \frac{π}{2}) = \frac{1}{3}

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

tg(3t + π/2) = \(± \frac{1}{\sqrt{3}}\)

3t + π/2 = π/6 + πn

3t = π/6 - π/2 + πn = -π/3 + πn

t = -π/9 + πn/3

3t + π/2 = -π/6 + πn

3t = -π/6 - π/2 + πn = -2π/3 + πn

t = -2π/9 + πn/3

Заменим -π/9 на 8π/9 и -2π/9 на 7π/9

3t + \(\frac{π}{2}\) = \(\frac{π}{6}\) + πn

3t = -\(\frac{π}{3}\) + πn

t = -\(\frac{π}{9}\) + \(\frac{π}{3}\)n

3t + \(\frac{π}{2}\) = -\(\frac{π}{6}\) + πn

3t = -\(\frac{2π}{3}\) + πn

t = -\(\frac{2π}{9}\) + \(\frac{π}{3}\)n

Если tg = \(\frac{1}{\sqrt{3}}\) , то 3t + π/2 = π/6 + πn,

Если tg = -\(\frac{1}{\sqrt{3}}\) , то 3t + π/2 = -π/6 + πn,

Решение: 3t + π/2 = π/6 + πn, 3t = π/6 - π/2 + πn = -π/3 + πn, t = -π/9 + πn/3.

Решение: 3t + π/2 = -π/6 + πn, 3t = -π/6 - π/2 + πn = -2π/3 + πn, t = -2π/9 + πn/3.

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