6*sin(x)^2-5*sin(x)-4=0

$$6 sin^{2}{left (x right )} – 5 sin{left (x right )} – 4 = 0$$

Дано уравнение
$$6 sin^{2}{left (x right )} – 5 sin{left (x right )} – 4 = 0$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (frac{4}{3} right )}$$
$$x_{1} = 2 pi n + {asin}{left (frac{4}{3} right )}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (- frac{1}{2} right )}$$
$$x_{2} = 2 pi n – frac{pi}{6}$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n + pi – {asin}{left (frac{4}{3} right )}$$
$$x_{3} = 2 pi n + pi – {asin}{left (frac{4}{3} right )}$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n – {asin}{left (- frac{1}{2} right )} + pi$$
$$x_{4} = 2 pi n + frac{7 pi}{6}$$

-pi
x1 = —-
6

$$x_{1} = – frac{pi}{6}$$

7*pi
x2 = —-
6

$$x_{2} = frac{7 pi}{6}$$

x3 = pi – re(asin(4/3)) – I*im(asin(4/3))

$$x_{3} = – Re{left({asin}{left (frac{4}{3} right )}right)} + pi – i Im{left({asin}{left (frac{4}{3} right )}right)}$$

x4 = I*im(asin(4/3)) + re(asin(4/3))

$$x_{4} = Re{left({asin}{left (frac{4}{3} right )}right)} + i Im{left({asin}{left (frac{4}{3} right )}right)}$$

x1 = -69.6386371546000

x2 = -103.148958793000

x3 = -88.4881930761000

x4 = -38.2227106187000

x5 = 100.007366139000

x6 = -78.0162175641000

x7 = -15.1843644924000

x8 = 41.3643032723000

x9 = -216.246294322000

x10 = 72.7802298082000

x11 = -40.3171057211000

x12 = 9.94837673637000

x13 = 62.3082542962000

x14 = 12.0427718388000

x15 = -21.4675497995000

x16 = -2.61799387799000

x17 = 49.7418836818000

x18 = -90.5825881785000

x19 = -46.6002910282000

x20 = -71.7330322570000

x21 = 47.6474885794000

x22 = 87.4409955249000

x23 = -63.3554518474000

x24 = -44.5058959259000

x25 = -27.7507351067000

x26 = 68.5914396034000

x27 = -1860.34644970000

x28 = 93.7241808321000

x29 = 5.75958653158000

x30 = -65.4498469498000

x31 = 43.4586983747000

x32 = -0.523598775598000

x33 = 35.0811179651000

x34 = 56.0250689890000

x35 = 22.5147473507000

x36 = -82.2050077689000

x37 = -84.2994028713000

x38 = 18.3259571459000

x39 = -57.0722665402000

x40 = -13.0899693900000

x41 = -145.036860841000

x42 = 85.3466004225000

x43 = -34.0339204139000

x44 = 30.8923277603000

x45 = 53.9306738866000

x46 = 66.4970445010000

x47 = -109.432144100000

x48 = 3.66519142919000

x49 = 131.423292675000

x50 = 24.6091424531000

x51 = 28.7979326579000

x52 = 97.9129710369000

x53 = 74.8746249106000

x54 = -59.1666616426000

x55 = 60.2138591938000

x56 = -19.3731546971000

x57 = 91.6297857297000

x58 = -31.9395253115000

x59 = -25.6563400043000

x60 = 16.2315620435000

x61 = -75.9218224618000