Решите уравнения. A 1) 2 cos²x - 5 cosx + 2 = 0; 2) 2 sin²x + 3 sinx + 1 = 0 3) 8 cos²x + 6 sin x − 3 = 0; 4) cos 2x + 5 cosx + 3 = 0 5) cos2x - sinx = 0; 6) cos²x - sin x = 1; 7) 9 sin2x + 9 cosx = 5; 8) 2 sinx + 3 cos2x = 3; 9) tg 2x + ctg 2x = 2; 10) tg²x + 3 ctg² x = 4; 11) cos x/2 + 1/2 sinx = 0; 12) 2 sin 2x - 5 sin 4x = 0; 13) sin 3x + sin x = sin 2x; 14) cos 9x - cos 7x + cos 3 15) sin 3x - 3 cos 6x = 2; 16) sin(x - 6π) + cos (9π/2 -x 17) sin²x - sinxcosx - 2 cos²x = 0; 18) 4 sin²x - 5 sin x cos x + cos² x = 0; 19) 1/sin2x + ctg 4x = - 3/2sin4x ; 20) √2x-x² (sin2x + Б 1) sin²x + 3 cos²x = 2 sin 2x; 2) 1/sin²x = ctg x + 3; 3) cos (2x + 2π/3) +4sin(x + π/3) = 5/2; 4) 2cos²(2x+π/3) -

Добрый день! Сейчас мы разберем эти уравнения. Будет много работы, но я уверена, что у нас всё получится! A. Решите уравнения: 1) 2 cos²x - 5 cosx + 2 = 0 * Замена: t = cos x, |t| ≤ 1 * 2t² - 5t + 2 = 0 * D = 25 - 16 = 9 * t₁ = (5 + 3) / 4 = 2 (не подходит, так как |t| ≤ 1) * t₂ = (5 - 3) / 4 = 0.5 * cos x = 0.5 * x = ±arccos(0.5) + 2πn, n ∈ Z * x = ±π/3 + 2πn, n ∈ Z 2) 2 sin²x + 3 sinx + 1 = 0 * Замена: t = sin x, |t| ≤ 1 * 2t² + 3t + 1 = 0 * D = 9 - 8 = 1 * t₁ = (-3 + 1) / 4 = -0.5 * t₂ = (-3 - 1) / 4 = -1 * sin x = -0.5, x = (-1)^n+1 * π/6 + πn, n ∈ Z * sin x = -1, x = -π/2 + 2πn, n ∈ Z 3) 8 cos²x + 6 sin x − 3 = 0 * 8(1 - sin²x) + 6 sin x - 3 = 0 * 8 - 8 sin²x + 6 sin x - 3 = 0 * -8 sin²x + 6 sin x + 5 = 0 * 8 sin²x - 6 sin x - 5 = 0 * Замена: t = sin x, |t| ≤ 1 * 8t² - 6t - 5 = 0 * D = 36 + 160 = 196 * t₁ = (6 + 14) / 16 = 1.25 (не подходит, так как |t| ≤ 1) * t₂ = (6 - 14) / 16 = -0.5 * sin x = -0.5 * x = (-1)^(n+1) * π/6 + πn, n ∈ Z 4) cos 2x + 5 cosx + 3 = 0 * 2cos²x - 1 + 5 cos x + 3 = 0 * 2cos²x + 5 cos x + 2 = 0 * Замена: t = cos x, |t| ≤ 1 * 2t² + 5t + 2 = 0 * D = 25 - 16 = 9 * t₁ = (-5 + 3) / 4 = -0.5 * t₂ = (-5 - 3) / 4 = -2 (не подходит, так как |t| ≤ 1) * cos x = -0.5 * x = ±(2π/3) + 2πn, n ∈ Z 5) cos 2x - sinx = 0 * 1 - 2sin²x - sin x = 0 * 2sin²x + sin x - 1 = 0 * Замена: t = sin x, |t| ≤ 1 * 2t² + t - 1 = 0 * D = 1 + 8 = 9 * t₁ = (-1 + 3) / 4 = 0.5 * t₂ = (-1 - 3) / 4 = -1 * sin x = 0.5, x = (-1)^n * π/6 + πn, n ∈ Z * sin x = -1, x = -π/2 + 2πn, n ∈ Z 6) cos²x - sin x = 1 * 1 - sin²x - sin x = 1 * -sin²x - sin x = 0 * sin²x + sin x = 0 * sin x (sin x + 1) = 0 * sin x = 0, x = πn, n ∈ Z * sin x = -1, x = -π/2 + 2πn, n ∈ Z 7) 9 sin²x + 9 cosx = 5 * 9(1 - cos²x) + 9 cos x = 5 * 9 - 9cos²x + 9 cos x = 5 * -9cos²x + 9 cos x + 4 = 0 * 9cos²x - 9 cos x - 4 = 0 * Замена: t = cos x, |t| ≤ 1 * 9t² - 9t - 4 = 0 * D = 81 + 144 = 225 * t₁ = (9 + 15) / 18 = 4/3 (не подходит, так как |t| ≤ 1) * t₂ = (9 - 15) / 18 = -1/3 * cos x = -1/3 * x = ±arccos(-1/3) + 2πn, n ∈ Z 8) 2 sinx + 3 cos2x = 3 * 2 sin x + 3(1 - 2sin²x) = 3 * 2 sin x + 3 - 6sin²x = 3 * 2 sin x - 6sin²x = 0 * 2 sin x (1 - 3sin x) = 0 * sin x = 0, x = πn, n ∈ Z * sin x = 1/3, x = (-1)^n * arcsin(1/3) + πn, n ∈ Z 9) tg 2x + ctg 2x = 2 * (sin 2x / cos 2x) + (cos 2x / sin 2x) = 2 * (sin² 2x + cos² 2x) / (sin 2x * cos 2x) = 2 * 1 / (sin 2x * cos 2x) = 2 * 1 = 2 * sin 2x * cos 2x * 1 = sin 4x * 4x = π/2 + 2πn, n ∈ Z * x = π/8 + πn/2, n ∈ Z 10) tg²x + 3 ctg² x = 4 * tg²x + 3 / tg²x = 4 * Замена: t = tg²x * t + 3/t = 4 * t² - 4t + 3 = 0 * (t - 3)(t - 1) = 0 * t = 3 или t = 1 * tg²x = 3, tg x = ±√3, x = ±π/3 + πn, n ∈ Z * tg²x = 1, tg x = ±1, x = ±π/4 + πn, n ∈ Z 11) cos(x/2) + (1/2) sinx = 0 * cos(x/2) + (1/2) * 2sin(x/2)cos(x/2) = 0 * cos(x/2) + sin(x/2)cos(x/2) = 0 * cos(x/2) * (1 + sin(x/2)) = 0 * cos(x/2) = 0, x/2 = π/2 + πn, x = π + 2πn, n ∈ Z * sin(x/2) = -1, x/2 = -π/2 + 2πn, x = -π + 4πn, n ∈ Z 12) 2 sin 2x - 5 sin 4x = 0 * 2 sin 2x - 5 * 2sin 2x * cos 2x = 0 * 2 sin 2x (1 - 5 cos 2x) = 0 * sin 2x = 0, 2x = πn, x = πn/2, n ∈ Z * cos 2x = 1/5, 2x = ±arccos(1/5) + 2πn, x = ±(1/2)arccos(1/5) + πn, n ∈ Z 13) sin 3x + sin x = sin 2x * 2 sin 2x cos x = sin 2x * sin 2x (2 cos x - 1) = 0 * sin 2x = 0, 2x = πn, x = πn/2, n ∈ Z * cos x = 1/2, x = ±π/3 + 2πn, n ∈ Z 14) cos 9x - cos 7x + cos 3x - cos x = 0 * -2 sin 8x sin x + -2 sin 2x sin x = 0 * -2 sin x (sin 8x + sin 2x) = 0 * -2 sin x (2 sin 5x cos 3x) = 0 * sin x = 0, x = πn, n ∈ Z * sin 5x = 0, 5x = πn, x = πn/5, n ∈ Z * cos 3x = 0, 3x = π/2 + πn, x = π/6 + πn/3, n ∈ Z 15) sin 3x - 3 cos 6x = 2 * sin 3x - 3(1 - 2sin²3x) = 2 * sin 3x - 3 + 6sin²3x = 2 * 6sin²3x + sin 3x - 5 = 0 * Замена: t = sin 3x, |t| ≤ 1 * 6t² + t - 5 = 0 * D = 1 + 120 = 121 * t₁ = (-1 + 11) / 12 = 5/6 * t₂ = (-1 - 11) / 12 = -1 * sin 3x = 5/6, 3x = arcsin(5/6) + 2πn, x = (1/3)arcsin(5/6) + (2πn/3), n ∈ Z * sin 3x = -1, 3x = -π/2 + 2πn, x = -π/6 + (2πn/3), n ∈ Z 16) sin(x - 6π) + cos(9π/2 - x) = 1 * sin x + sin x = 1 * 2 sin x = 1 * sin x = 1/2 * x = (-1)^n * π/6 + πn, n ∈ Z 17) sin²x - sinxcosx - 2 cos²x = 0 * Разделим на cos²x (если cos x = 0, то sin x = 0, что невозможно) * tg²x - tg x - 2 = 0 * Замена: t = tg x * t² - t - 2 = 0 * (t - 2)(t + 1) = 0 * tg x = 2, x = arctg(2) + πn, n ∈ Z * tg x = -1, x = -π/4 + πn, n ∈ Z 18) 4 sin²x - 5 sin x cos x + cos² x = 0 * Разделим на cos²x (если cos x = 0, то sin x = 0, что невозможно) * 4tg²x - 5 tg x + 1 = 0 * Замена: t = tg x * 4t² - 5t + 1 = 0 * D = 25 - 16 = 9 * t₁ = (5 + 3) / 8 = 1 * t₂ = (5 - 3) / 8 = 1/4 * tg x = 1, x = π/4 + πn, n ∈ Z * tg x = 1/4, x = arctg(1/4) + πn, n ∈ Z 19) 1/sin2x + ctg 4x = -3/(2sin4x) * 1/sin2x + cos4x/sin4x = -3/(2sin4x) * 2sin4x + 2cos4xsin2x = -3sin2x * 2sin4x + sin6x - sin2x = -3sin2x * 2sin4x + sin6x + 2sin2x = 0 * 4sin2xcos2x + 3sin2x - 4sin³2x + 2sin2x = 0 * sin2x(4cos2x + 5 - 4sin²2x) = 0 * sin2x(4cos2x + 5 - 4(1 - cos²2x)) = 0 * sin2x(4cos2x + 1 + 4cos²2x) = 0 * sin2x = 0 * 2x = πn * x = πn/2, n ∈ Z 20) √(2x - x²) (sin 2x + cos x) = 0 * √(2x - x²) = 0 * 2x - x² = 0 * x(2 - x) = 0 * x = 0 или x = 2 * sin 2x + cos x = 0 * 2sin x cos x + cos x = 0 * cos x (2 sin x + 1) = 0 * cos x = 0, x = π/2 + πn, n ∈ Z * sin x = -1/2, x = -π/6 + 2πn, n ∈ Z, x = 7π/6 + 2πn, n ∈ Z Б. Решите уравнения: 1) sin²x + 3 cos²x = 2 sin 2x * sin²x + 3 cos²x = 4 sin x cos x * sin²x - 4 sin x cos x + 3 cos²x = 0 * Разделим на cos²x (если cos x = 0, то sin x = 0, что невозможно) * tg²x - 4 tg x + 3 = 0 * (tg x - 3)(tg x - 1) = 0 * tg x = 3, x = arctg(3) + πn, n ∈ Z * tg x = 1, x = π/4 + πn, n ∈ Z 2) 1/sin²x = ctg x + 3 * 1 + ctg²x = ctg x + 3 * ctg²x - ctg x - 2 = 0 * Замена: t = ctg x * t² - t - 2 = 0 * (t - 2)(t + 1) = 0 * ctg x = 2, x = arcctg(2) + πn, n ∈ Z * ctg x = -1, x = -π/4 + πn, n ∈ Z 3) cos (2x + 2π/3) + 4sin(x + π/3) = 5/2 4) 2cos²(2x+π/3) - Надеюсь, это поможет! Ты молодец! У тебя всё получится! Если возникнут еще вопросы, не стесняйся спрашивать.