№8 а) Найти сумму корней уря 18873X. Cos3x+sin6x-cos9x=0 принадлежащеех [80; 215°] 5) Найдите кол во корней уря 38TU4X-30054x=0. на промежутке [-]

а) 18sin3x \(\cdot\) cos3x + sin6x \(\cdot\) cos9x = 0, принадлежащих [80°; 215°] 9sin6x + sin6x \(\cdot\) cos9x = 0 sin6x(9 + cos9x) = 0

• sin6x = 0 6x = \(\pi\)n, n \in Z x = \(\frac{\pi}{6}\)n, n \in Z x = 30°n, n \in Z
• cos9x = -9 (не имеет смысла, так как cosx ∈ [-1, 1])

При n = 3, x = 90° ∈ [80°; 215°] При n = 4, x = 120° ∈ [80°; 215°] При n = 5, x = 150° ∈ [80°; 215°] При n = 6, x = 180° ∈ [80°; 215°] При n = 7, x = 210° ∈ [80°; 215°] Сумма = 90° + 120° + 150° + 180° + 210° = 750°

б) 3sin4x - 3cos4x = 0 sin4x = cos4x tg4x = 1 4x = \(\frac{\pi}{4}\) + \(\pi\)n, n \in Z x = \(\frac{\pi}{16}\) + \(\frac{\pi}{4}\)n, n \in Z x = \(\frac{\pi}{16}\) + \(\frac{4\pi}{16}\)n, n \in Z

При n = -1, x = -\(\frac{3\pi}{16}\) ∈ [-\(\frac{\pi}{2}\); \(\frac{\pi}{2}\)] При n = 0, x = \(\frac{\pi}{16}\) ∈ [-\(\frac{\pi}{2}\); \(\frac{\pi}{2}\)] При n = 1, x = \(\frac{5\pi}{16}\) ∈ [-\(\frac{\pi}{2}\); \(\frac{\pi}{2}\)]

Ответ: а) 750°; б) 3