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

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

Дано уравнение
$$5 sin{left (x right )} + 2 cos^{2}{left (x right )} – 4 = 0$$
преобразуем
$$5 sin{left (x right )} + cos{left (2 x right )} – 3 = 0$$
$$- 2 sin^{2}{left (x right )} + 5 sin{left (x right )} – 2 = 0$$
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.
$$a = -2$$
$$b = 5$$
$$c = -2$$
, то

D = b^2 – 4 * a * c =

(5)^2 – 4 * (-2) * (-2) = 9

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

pi
x1 = —
6

$$x_{1} = frac{pi}{6}$$

/ ___
x2 = 2*atan2 + / 3 /

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

/ / ___ / / ___
| |1 I*/ 3 || | |1 I*/ 3 ||
x3 = 2*re|atan|- – ——-|| + 2*I*im|atan|- – ——-||
2 2 // 2 2 //

$$x_{3} = 2 Re{left({atan}{left (frac{1}{2} – frac{sqrt{3} i}{2} right )}right)} + 2 i Im{left({atan}{left (frac{1}{2} – frac{sqrt{3} i}{2} right )}right)}$$

/ / ___ / / ___
| |1 I*/ 3 || | |1 I*/ 3 ||
x4 = 2*re|atan|- + ——-|| + 2*I*im|atan|- + ——-||
2 2 // 2 2 //

$$x_{4} = 2 Re{left({atan}{left (frac{1}{2} + frac{sqrt{3} i}{2} right )}right)} + 2 i Im{left({atan}{left (frac{1}{2} + frac{sqrt{3} i}{2} right )}right)}$$

x1 = -35.0811179651000

x2 = -81.1578102177000

x3 = -30.8923277603000

x4 = -66.4970445010000

x5 = -49.7418836818000

x6 = -28.7979326579000

x7 = 2.61799387799000

x8 = 71.7330322570000

x9 = 6.80678408278000

x10 = 84.2994028713000

x11 = 19.3731546971000

x12 = 69.6386371546000

x13 = 88.4881930761000

x14 = 59.1666616426000

x15 = -68.5914396034000

x16 = -93.7241808321000

x17 = -100.007366139000

x18 = -22.5147473507000

x19 = 46.6002910282000

x20 = -91.6297857297000

x21 = 38.2227106187000

x22 = -60.2138591938000

x23 = 90.5825881785000

x24 = 44.5058959259000

x25 = -43.4586983747000

x26 = 40.3171057211000

x27 = 94.7713783833000

x28 = 50.7890812330000

x29 = -62.3082542962000

x30 = -85.3466004225000

x31 = -204.727121259000

x32 = -79.0634151153000

x33 = 34.0339204139000

x34 = -87.4409955249000

x35 = 101.054563690000

x36 = -74.8746249106000

x37 = -12.0427718388000

x38 = 52.8834763354000

x39 = -2190.21367833000

x40 = 534.594349886000

x41 = 7056.54069874000

x42 = 21.4675497995000

x43 = -41.3643032723000

x44 = 13.0899693900000

x45 = 31.9395253115000

x46 = -56.0250689890000

x47 = -18.3259571459000

x48 = 75.9218224618000

x49 = -3.66519142919000

x50 = -24.6091424531000

x51 = -16.2315620435000

x52 = 25.6563400043000

x53 = -5.75958653158000

x54 = 63.3554518474000

x55 = 15.1843644924000

x56 = 57.0722665402000

x57 = -97.9129710369000

x58 = -53.9306738866000

x59 = 0.523598775598000

x60 = 2113.76825709000

x61 = -37.1755130675000

x62 = 804.771318095000

x63 = 78.0162175641000

x64 = 65.4498469498000

x65 = -72.7802298082000

x66 = 27.7507351067000

x67 = 8.90117918517000

x68 = -47.6474885794000

x69 = 96.8657734857000

x70 = -9.94837673637000

x71 = 82.2050077689000