4*sin(x)^(3)+1=4*sin(x)^(2)+sin(x)

$$4 sin^{3}{left (x right )} + 1 = 4 sin^{2}{left (x right )} + sin{left (x right )}$$

Дано уравнение
$$4 sin^{3}{left (x right )} + 1 = 4 sin^{2}{left (x right )} + sin{left (x right )}$$
преобразуем
$$4 sin^{3}{left (x right )} – 4 sin^{2}{left (x right )} – sin{left (x right )} + 1 = 0$$
$$- 4 sin^{2}{left (x right )} – sin{left (x right )} + 4 sin^{3}{left (x right )} + 1 = 0$$
Сделаем замену
$$w = sin{left (x right )}$$
Дано уравнение:
$$4 w^{3} – 4 w^{2} – w + 1 = 0$$
преобразуем
$$- w + – 4 w^{2} + 4 w^{3} – 4 + 4 + 1 = 0$$
или
$$- w + – 4 w^{2} + 4 w^{3} – 4 – -4 + 1 = 0$$
$$- w – 1 + – 4 left(w^{2} – 1right) + 4 left(w^{3} – 1right) = 0$$
$$- w – 1 + – 4 left(w – 1right) left(w + 1right) + 4 left(w – 1right) left(w^{2} + w + 1^{2}right) = 0$$
Вынесем общий множитель -1 + w за скобки
получим:
$$left(w – 1right) left(- 4 left(w + 1right) + 4 left(w^{2} + w + 1^{2}right) – 1right) = 0$$
или
$$left(w – 1right) left(4 w^{2} – 1right) = 0$$
тогда:
$$w_{1} = 1$$
и также
получаем ур-ние
$$4 w^{2} – 1 = 0$$
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.
$$a = 4$$
$$b = 0$$
$$c = -1$$
, то

D = b^2 – 4 * a * c =

(0)^2 – 4 * (4) * (-1) = 16

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

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

w3 = (-b – sqrt(D)) / (2*a)

или
$$w_{2} = frac{1}{2}$$
$$w_{3} = – frac{1}{2}$$
Получаем окончательный ответ для 4*sin(x)^3 + 1 – 4*sin(x)^2 – sin(x) = 0:
$$w_{1} = 1$$
$$w_{2} = frac{1}{2}$$
$$w_{3} = – frac{1}{2}$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (1 right )}$$
$$x_{1} = 2 pi n + frac{pi}{2}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (frac{1}{2} right )}$$
$$x_{2} = 2 pi n + frac{pi}{6}$$
$$x_{3} = 2 pi n + {asin}{left (w_{3} right )}$$
$$x_{3} = 2 pi n + {asin}{left (- frac{1}{2} right )}$$
$$x_{3} = 2 pi n – frac{pi}{6}$$
$$x_{4} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{4} = 2 pi n – {asin}{left (1 right )} + pi$$
$$x_{4} = 2 pi n + frac{pi}{2}$$
$$x_{5} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{5} = 2 pi n – {asin}{left (frac{1}{2} right )} + pi$$
$$x_{5} = 2 pi n + frac{5 pi}{6}$$
$$x_{6} = 2 pi n – {asin}{left (w_{3} right )} + pi$$
$$x_{6} = 2 pi n – {asin}{left (- frac{1}{2} right )} + pi$$
$$x_{6} = 2 pi n + frac{7 pi}{6}$$

-pi
x1 = —-
6

$$x_{1} = – frac{pi}{6}$$

pi
x2 = —
6

$$x_{2} = frac{pi}{6}$$

pi
x3 = —
2

$$x_{3} = frac{pi}{2}$$

5*pi
x4 = —-
6

$$x_{4} = frac{5 pi}{6}$$

7*pi
x5 = —-
6

$$x_{5} = frac{7 pi}{6}$$

x1 = 58.1194642274000

x2 = -100.007366139000

x3 = 71.7330322570000

x4 = 41.3643032723000

x5 = 88.4881930761000

x6 = 20.4203521593000

x7 = -69.6386371546000

x8 = -5.75958653158000

x9 = -60.2138591938000

x10 = 89.5353903729000

x11 = -43.4586983747000

x12 = -71.7330322570000

x13 = 47.6474885794000

x14 = 89.5353905748000

x15 = 34.0339204139000

x16 = 95.8185760401000

x17 = -12.0427718388000

x18 = 21.4675497995000

x19 = 31.9395253115000

x20 = 18.3259571459000

x21 = -13.0899693900000

x22 = -47.6474885794000

x23 = 0.523598775598000

x24 = -31.9395253115000

x25 = -25.6563400043000

x26 = -75.9218224618000

x27 = -38.2227106187000

x28 = -23.5619449966000

x29 = -40.3171057211000

x30 = 62.3082542962000

x31 = 14.1371670702000

x32 = 7.85398172873000

x33 = -15800.6402513000

x34 = 5.75958653158000

x35 = -80.1106125862000

x36 = -86.3937979377000

x37 = 66.4970445010000

x38 = 97.9129710369000

x39 = 75.9218224618000

x40 = -16.2315620435000

x41 = 12.0427718388000

x42 = 1.57079624161000

x43 = -34.0339204139000

x44 = -97.9129710369000

x45 = 91.6297857297000

x46 = -2.61799387799000

x47 = 78.0162175641000

x48 = -91.6297857297000

x49 = -67.5442421518000

x50 = 16.2315620435000

x51 = 100.007366139000

x52 = 84.2994028713000

x53 = -49.7418836818000

x54 = -93.7241808321000

x55 = -105.243354121000

x56 = -42.4115008333000

x57 = 9.94837673637000

x58 = -90.5825881785000

x59 = -46.6002910282000

x60 = -63.3554518474000

x61 = -87.4409955249000

x62 = 51.8362788848000

x63 = 68.5914396034000

x64 = -56.0250689890000

x65 = 56.0250689890000

x66 = -82.2050077689000

x67 = -757.647428291000

x68 = -57.0722665402000

x69 = 85.3466004225000

x70 = 30.8923277603000

x71 = 53.9306738866000

x72 = 24.6091424531000

x73 = -9.94837673637000

x74 = 82.2050077689000

x75 = 74.8746249106000

x76 = 45.5530932702000

x77 = 44.5058959259000

x78 = -78.0162175641000

x79 = -23.5619448325000

x80 = 38.2227106187000

x81 = 70.6858348307000

x82 = 40.3171057211000

x83 = -36.1283154283000

x84 = -84.2994028713000

x85 = -53.9306738866000

x86 = 102.101761340000

x87 = 26.7035378190000

x88 = -3.66519142919000

x89 = -29.8451301005000

x90 = 64.4026493159000

x91 = -27.7507351067000

x92 = 3.66519142919000

x93 = 27.7507351067000

x94 = 60.2138591938000

x95 = -19.3731546971000

x96 = -73.8274272809000