2*sin(x)^(3)-cos(2*x)-sin(x)=0

$$2 sin^{3}{left (x right )} – cos{left (2 x right )} – sin{left (x right )} = 0$$

Дано уравнение
$$2 sin^{3}{left (x right )} – cos{left (2 x right )} – sin{left (x right )} = 0$$
преобразуем
$$2 sin^{3}{left (x right )} + 2 sin^{2}{left (x right )} – sin{left (x right )} – 1 = 0$$
$$2 sin^{3}{left (x right )} + 2 sin^{2}{left (x right )} – sin{left (x right )} – 1 = 0$$
Сделаем замену
$$w = sin{left (x right )}$$
Дано уравнение:
$$2 w^{3} + 2 w^{2} – w – 1 = 0$$
преобразуем
$$- w + 2 w^{2} + 2 w^{3} + 2 – 2 – 1 = 0$$
или
$$- w + 2 w^{2} + 2 w^{3} – -2 – 2 – 1 = 0$$
$$- w + 1 + 2 left(w^{2} – 1right) + 2 left(w^{3} – -1right) = 0$$
$$- w + 1 + left(w – 1right) 2 left(w + 1right) + 2 left(w + 1right) left(w^{2} – w + left(-1right)^{2}right) = 0$$
Вынесем общий множитель 1 + w за скобки
получим:
$$left(w + 1right) left(2 left(w – 1right) + 2 left(w^{2} – w + left(-1right)^{2}right) – 1right) = 0$$
или
$$left(w + 1right) left(2 w^{2} – 1right) = 0$$
тогда:
$$w_{1} = -1$$
и также
получаем ур-ние
$$2 w^{2} – 1 = 0$$
Это уравнение вида

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

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$w_{2} = frac{sqrt{D} – b}{2 a}$$
$$w_{3} = frac{- sqrt{D} – b}{2 a}$$
где D = b^2 – 4*a*c – это дискриминант.
Т.к.
$$a = 2$$
$$b = 0$$
$$c = -1$$
, то

D = b^2 – 4 * a * c =

(0)^2 – 4 * (2) * (-1) = 8

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

w2 = (-b + sqrt(D)) / (2*a)

w3 = (-b – sqrt(D)) / (2*a)

или
$$w_{2} = frac{sqrt{2}}{2}$$
$$w_{3} = – frac{sqrt{2}}{2}$$
Получаем окончательный ответ для -1 – sin(x) + 2*sin(x)^2 + 2*sin(x)^3 = 0:
$$w_{1} = -1$$
$$w_{2} = frac{sqrt{2}}{2}$$
$$w_{3} = – frac{sqrt{2}}{2}$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (-1 right )}$$
$$x_{1} = 2 pi n – frac{pi}{2}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (frac{sqrt{2}}{2} right )}$$
$$x_{2} = 2 pi n + frac{pi}{4}$$
$$x_{3} = 2 pi n + {asin}{left (w_{3} right )}$$
$$x_{3} = 2 pi n + {asin}{left (- frac{sqrt{2}}{2} right )}$$
$$x_{3} = 2 pi n – frac{pi}{4}$$
$$x_{4} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{4} = 2 pi n – {asin}{left (-1 right )} + pi$$
$$x_{4} = 2 pi n + frac{3 pi}{2}$$
$$x_{5} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{5} = 2 pi n – {asin}{left (frac{sqrt{2}}{2} right )} + pi$$
$$x_{5} = 2 pi n + frac{3 pi}{4}$$
$$x_{6} = 2 pi n – {asin}{left (w_{3} right )} + pi$$
$$x_{6} = 2 pi n – {asin}{left (- frac{sqrt{2}}{2} right )} + pi$$
$$x_{6} = 2 pi n + frac{5 pi}{4}$$

-pi
x1 = —-
2

$$x_{1} = – frac{pi}{2}$$

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

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

3*pi /log(2) / ___
x3 = —- + I*|—— – log/ 2 /|
4 2 /

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

pi /log(2) / ___
x4 = – — + I*|—— – log/ 2 /|
4 2 /

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

pi /log(2) / ___
x5 = — + I*|—— – log/ 2 /|
4 2 /

$$x_{5} = frac{pi}{4} + i left(- log{left (sqrt{2} right )} + frac{1}{2} log{left (2 right )}right)$$

x1 = -77.7544181763000

x2 = 90.3207887907000

x3 = 22.7765467385000

x4 = -93.4623814443000

x5 = 77.7544181763000

x6 = -7.85398151564000

x7 = -71.4712328692000

x8 = -45.5530935610000

x9 = 33.7721210261000

x10 = -47.9092879672000

x11 = 66.7588438888000

x12 = 69.9004365424000

x13 = -89.5353908032000

x14 = -25.9181393921000

x15 = 63.6172512352000

x16 = 30.6305283725000

x17 = -49.4800842940000

x18 = -14.1371668576000

x19 = 84.0376034835000

x20 = 36.1283156124000

x21 = 54.1924732744000

x22 = 2.35619449019000

x23 = -33.7721210261000

x24 = 10.2101761242000

x25 = 62.0464549084000

x26 = -89.5353907108000

x27 = -58.1194640130000

x28 = 76.1836218496000

x29 = 49.4800842940000

x30 = -2.35619449019000

x31 = -5.49778714378000

x32 = -55.7632696012000

x33 = 60.4756585816000

x34 = -54.1924732744000

x35 = -38.4845100065000

x36 = 40.0553063333000

x37 = -32.2013246993000

x38 = -79.3252145031000

x39 = -18.0641577581000

x40 = -62.0464549084000

x41 = 46.3384916404000

x42 = -11.7809724510000

x43 = -64.4026492015000

x44 = 80.1106110912000

x45 = -65.1880475620000

x46 = -45.5530929409000

x47 = 85.6083998103000

x48 = -26.7035359875000

x49 = 32.2013246993000

x50 = -76.1836218496000

x51 = 18.0641577581000

x52 = -99.7455667515000

x53 = -69.9004365424000

x54 = 3.92699081699000

x55 = 80.8960108299000

x56 = 93.4623814443000

x57 = 11.7809724510000

x58 = 98.1747704247000

x59 = -19.6349540849000

x60 = 38.4845100065000

x61 = -209.701309627000

x62 = -21.2057504117000

x63 = 24.3473430653000

x64 = -84.0376034835000

x65 = -35.3429173529000

x66 = -41.6261026601000

x67 = -91.8915851175000

x68 = 82.4668071567000

x69 = 96.6039740979000

x70 = 25.9181393921000

x71 = 42.4115007446000

x72 = -95.8185758687000

x73 = -76.9690200451000

x74 = -8.63937979737000

x75 = -82.4668071567000

x76 = 86.3937978996000

x77 = -10.2101761242000

x78 = -40.0553063333000

x79 = -85.6083998103000

x80 = 73.8274274488000

x81 = -27.4889357189000

x82 = -98.1747704247000

x83 = 47.9092879672000

x84 = 16.4933614313000

x85 = 29.8451302963000

x86 = -3.92699081699000

x87 = 68.3296402156000

x88 = 19.6349540849000

x89 = -1.57079640899000

x90 = 5.49778714378000

x91 = 99.7455667515000

x92 = 52.6216769476000

x93 = -68.3296402156000

x94 = 55.7632696012000

x95 = -24.3473430653000

x96 = 8.63937979737000

x97 = -51.8362786930000