3*cos(2*x)-5*sin(x)+1=0

$$- 5 sin{left (x right )} + 3 cos{left (2 x right )} + 1 = 0$$

Дано уравнение
$$- 5 sin{left (x right )} + 3 cos{left (2 x right )} + 1 = 0$$
преобразуем
$$- 6 sin^{2}{left (x right )} – 5 sin{left (x right )} + 4 = 0$$
$$- 6 sin^{2}{left (x right )} – 5 sin{left (x right )} + 4 = 0$$
Сделаем замену
$$w = sin{left (x right )}$$
Это уравнение вида

a*w^2 + b*w + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$w_{1} = frac{sqrt{D} – b}{2 a}$$
$$w_{2} = frac{- sqrt{D} – b}{2 a}$$
где D = b^2 – 4*a*c – это дискриминант.
Т.к.
$$a = -6$$
$$b = -5$$
$$c = 4$$
, то

D = b^2 – 4 * a * c =

(-5)^2 – 4 * (-6) * (4) = 121

Т.к. D > 0, то уравнение имеет два корня.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = – frac{4}{3}$$
$$w_{2} = frac{1}{2}$$
делаем обратную замену
$$sin{left (x right )} = w$$
Дано уравнение
$$sin{left (x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 pi n + {asin}{left (w right )}$$
$$x = 2 pi n – {asin}{left (w right )} + pi$$
Или
$$x = 2 pi n + {asin}{left (w right )}$$
$$x = 2 pi n – {asin}{left (w right )} + pi$$
, где n – любое целое число
подставляем w:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (- frac{4}{3} right )}$$
$$x_{1} = 2 pi n – {asin}{left (frac{4}{3} right )}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (frac{1}{2} right )}$$
$$x_{2} = 2 pi n + frac{pi}{6}$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n + pi – {asin}{left (- frac{4}{3} right )}$$
$$x_{3} = 2 pi n + pi + {asin}{left (frac{4}{3} right )}$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n – {asin}{left (frac{1}{2} right )} + pi$$
$$x_{4} = 2 pi n + frac{5 pi}{6}$$

$$x_{1} = – frac{pi}{2} – i log{left (left|{- frac{sqrt{7} i}{3} + frac{4 i}{3}}right| right )}$$

/| ___|
pi ||4*I I*/ 7 ||
x2 = – — – I*log||— + ——-||
2 | 3 3 |/

$$x_{2} = – frac{pi}{2} – i log{left (left|{frac{sqrt{7} i}{3} + frac{4 i}{3}}right| right )}$$

/ ___
|I / 3 |
x3 = -I*log|- – —–|
2 2 /

$$x_{3} = – i log{left (- frac{sqrt{3}}{2} + frac{i}{2} right )}$$

/ ___
|I / 3 |
x4 = -I*log|- + —–|
2 2 /

$$x_{4} = – i log{left (frac{sqrt{3}}{2} + frac{i}{2} right )}$$

x1 = -35.0811179651000

x2 = -81.1578102177000

x3 = -100.007366139000

x4 = -66.4970445010000

x5 = -110.479341651000

x6 = -49.7418836818000

x7 = -62.3082542962000

x8 = 2.61799387799000

x9 = 71.7330322570000

x10 = 84.2994028713000

x11 = 19.3731546971000

x12 = 69.6386371546000

x13 = 88.4881930761000

x14 = 6.80678408278000

x15 = 82.2050077689000

x16 = -93.7241808321000

x17 = -22.5147473507000

x18 = 46.6002910282000

x19 = 38.2227106187000

x20 = -60.2138591938000

x21 = 44.5058959259000

x22 = -43.4586983747000

x23 = 40.3171057211000

x24 = 94.7713783833000

x25 = -85.3466004225000

x26 = -79.0634151153000

x27 = 34.0339204139000

x28 = -87.4409955249000

x29 = -53.9306738866000

x30 = -12.0427718388000

x31 = 52.8834763354000

x32 = 50.7890812330000

x33 = -68.5914396034000

x34 = 21.4675497995000

x35 = -41.3643032723000

x36 = 31.9395253115000

x37 = 90.5825881785000

x38 = -56.0250689890000

x39 = -18.3259571459000

x40 = 75.9218224618000

x41 = -3.66519142919000

x42 = -24.6091424531000

x43 = -16.2315620435000

x44 = 25.6563400043000

x45 = -5.75958653158000

x46 = 63.3554518474000

x47 = -1203.75358510000

x48 = 115.715329407000

x49 = -97.9129710369000

x50 = -47.6474885794000

x51 = 0.523598775598000

x52 = -37.1755130675000

x53 = 78.0162175641000

x54 = 65.4498469498000

x55 = -72.7802298082000

x56 = 27.7507351067000

x57 = 8.90117918517000

x58 = 96.8657734857000

x59 = -9.94837673637000

x60 = -91.6297857297000