9. Вычислите: a) cos 0,3π sin 0,2π + sin 0,3π cos 0,2π б) (tg(π/10) + tg(3π/20)) / (1 - tg(π/10) * tg(3π/20)) в) sin(5π/18) cos(π/9) - sin(π/9) cos(5π/18) г) (sin(5π/12) sin(7π/12)) / (sin(5π/12) cos(7π/12))

a) Используем формулу синуса суммы: sin(a+b) = sin(a)cos(b) + cos(a)sin(b). Тогда cos 0,3π sin 0,2π + sin 0,3π cos 0,2π = sin(0,3π + 0,2π) = sin(0,5π) = sin(π/2) = 1. б) Используем формулу тангенса суммы: tg(a+b) = (tg(a) + tg(b)) / (1 - tg(a)tg(b)). Тогда (tg(π/10) + tg(3π/20)) / (1 - tg(π/10) * tg(3π/20)) = tg(π/10 + 3π/20) = tg(2π/20 + 3π/20) = tg(5π/20) = tg(π/4) = 1. в) Используем формулу синуса разности: sin(a-b) = sin(a)cos(b) - cos(a)sin(b). Тогда sin(5π/18) cos(π/9) - sin(π/9) cos(5π/18) = sin(5π/18 - π/9) = sin(5π/18 - 2π/18) = sin(3π/18) = sin(π/6) = 1/2. г) (sin(5π/12) sin(7π/12)) / (sin(5π/12) cos(7π/12)) = sin(5π/12) / sin(5π/12) * sin(7π/12) / cos(7π/12) = tg(7π/12). Чтобы вычислить tg(7π/12), заметим, что 7π/12 = π/3 + π/4. Тогда tg(7π/12) = tg(π/3 + π/4) = (tg(π/3) + tg(π/4)) / (1 - tg(π/3)tg(π/4)) = (√3 + 1) / (1 - √3) = ((√3 + 1) * (1+√3)) / ((1-√3) * (1+√3)) = (1 + 2√3 + 3) / (1 - 3) = (4 + 2√3) / (-2) = -2 - √3.