Алгебра.Однородные тригонометрические уравнения. Метод введения вспомогательного аргумента. Универсальная подстановка

1. sin x + 4 cos x sin x - cos x = -1
sin x - cos x + 2sin 2x = -1
Перенесем -1 влево и домножим на √2
√2 (1/√2 sin x - 1/√2 cos x) + 2sin 2x + 1 = 0
√2 sin(x - π/4) + 2sin 2x + 1 = 0
Если x = 3π/4, то √2 sin(π/2) + 2sin (3π/2) + 1 = √2 - 2 + 1 ≠ 0


2. 5cos²3x - 2 + 3sin²3x - 2√3 sin 3x cos 3x = 0
5cos²3x + 3sin²3x - √3 sin 6x - 2 = 0
5cos²3x + 3(1-cos²3x) - √3 sin 6x - 2 = 0
2cos²3x - √3 sin 6x + 1 = 0
2cos²3x - 2√3 sin 3x cos 3x + 1 = 0
2(cos 3x - √3/2 sin 3x)^2 - 1/2 sin^2 3x + 1 = 0
2(cos 3x sin(-π/3) + sin 3x cos(-π/3))^2 - 1/2 (1 - cos 6x)/2 + 1 = 0


3. 3cos²2x + 2sin²2x = 2.5 sin 4x
3cos²2x + 2(1 - cos²2x) = 2.5 * 2sin 2x cos 2x
cos²2x + 2 = 5sin 2x cos 2x
1 + cos 4x + 4 = 10sin 2x cos 2x
1 + cos 4x + 4 = 5sin 4x


4. 6sin²(x/2 - π/6) + 0.5 sin(x - π/3) = 2 + cos²(π/6 - x/2)
6sin²(x/2 - π/6) + 0.5 sin(x - π/3) = 2 + sin²(x/2 - π/6)
5sin²(x/2 - π/6) + 0.5 sin(x - π/3) - 2 = 0
5sin²(x/2 - π/6) + sin(x/2 - π/6)cos(x/2 - π/6) - 2 = 0