8*cos(x)^(2)+6*sin(x)-3=0

$$6 sin{left (x right )} + 8 cos^{2}{left (x right )} – 3 = 0$$

Дано уравнение
$$6 sin{left (x right )} + 8 cos^{2}{left (x right )} – 3 = 0$$
преобразуем
$$- 8 sin^{2}{left (x right )} + 6 sin{left (x right )} + 5 = 0$$
$$- 8 sin^{2}{left (x right )} + 6 sin{left (x right )} + 5 = 0$$
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.
$$a = -8$$
$$b = 6$$
$$c = 5$$
, то

D = b^2 – 4 * a * c =

(6)^2 – 4 * (-8) * (5) = 196

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = – frac{1}{2}$$
$$w_{2} = frac{5}{4}$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (- frac{1}{2} right )}$$
$$x_{1} = 2 pi n – frac{pi}{6}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (frac{5}{4} right )}$$
$$x_{2} = 2 pi n + {asin}{left (frac{5}{4} right )}$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n – {asin}{left (- frac{1}{2} right )} + pi$$
$$x_{3} = 2 pi n + frac{7 pi}{6}$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n + pi – {asin}{left (frac{5}{4} right )}$$
$$x_{4} = 2 pi n + pi – {asin}{left (frac{5}{4} right )}$$

-pi
x1 = —-
6

$$x_{1} = – frac{pi}{6}$$

/ ___
x2 = -2*atan2 + / 3 /

$$x_{2} = – 2 {atan}{left (sqrt{3} + 2 right )}$$

/ /4 3*I / /4 3*I
x3 = 2*re|atan|- – —|| + 2*I*im|atan|- – —||
5 5 // 5 5 //

$$x_{3} = 2 Re{left({atan}{left (frac{4}{5} – frac{3 i}{5} right )}right)} + 2 i Im{left({atan}{left (frac{4}{5} – frac{3 i}{5} right )}right)}$$

/ /4 3*I / /4 3*I
x4 = 2*re|atan|- + —|| + 2*I*im|atan|- + —||
5 5 // 5 5 //

$$x_{4} = 2 Re{left({atan}{left (frac{4}{5} + frac{3 i}{5} right )}right)} + 2 i Im{left({atan}{left (frac{4}{5} + frac{3 i}{5} right )}right)}$$

x1 = 93.7241808321000

x2 = -69.6386371546000

x3 = -103.148958793000

x4 = -88.4881930761000

x5 = -38.2227106187000

x6 = 100.007366139000

x7 = -78.0162175641000

x8 = -18102.3804688000

x9 = 37.1755130675000

x10 = -22940.4331553000

x11 = 41.3643032723000

x12 = 72.7802298082000

x13 = -40.3171057211000

x14 = 9.94837673637000

x15 = 62.3082542962000

x16 = 12.0427718388000

x17 = -21.4675497995000

x18 = -2.61799387799000

x19 = 49.7418836818000

x20 = -90.5825881785000

x21 = -46.6002910282000

x22 = -71.7330322570000

x23 = 47.6474885794000

x24 = 87.4409955249000

x25 = -63.3554518474000

x26 = -44.5058959259000

x27 = -27.7507351067000

x28 = 68.5914396034000

x29 = -15.1843644924000

x30 = 5.75958653158000

x31 = 21071.1855264000

x32 = -50.7890812330000

x33 = -65.4498469498000

x34 = 43.4586983747000

x35 = -113.620934305000

x36 = -0.523598775598000

x37 = 56.0250689890000

x38 = 22.5147473507000

x39 = -82.2050077689000

x40 = -84.2994028713000

x41 = 18.3259571459000

x42 = -57.0722665402000

x43 = -13.0899693900000

x44 = 85.3466004225000

x45 = -34.0339204139000

x46 = 30.8923277603000

x47 = -101.054563690000

x48 = 53.9306738866000

x49 = 66.4970445010000

x50 = 3.66519142919000

x51 = 24.6091424531000

x52 = 28.7979326579000

x53 = -6.80678408278000

x54 = 97.9129710369000

x55 = 74.8746249106000

x56 = -59.1666616426000

x57 = 60.2138591938000

x58 = -19.3731546971000

x59 = 91.6297857297000

x60 = -31.9395253115000

x61 = -25.6563400043000

x62 = -1024.68280385000

x63 = 16.2315620435000

x64 = -75.9218224618000