2*sin(x)^(2)+5*cos(x)+1=0

$$2 sin^{2}{left (x right )} + 5 cos{left (x right )} + 1 = 0$$

Дано уравнение
$$2 sin^{2}{left (x right )} + 5 cos{left (x right )} + 1 = 0$$
преобразуем
$$5 cos{left (x right )} – cos{left (2 x right )} + 2 = 0$$
$$- 2 cos^{2}{left (x right )} + 5 cos{left (x right )} + 3 = 0$$
Сделаем замену
$$w = cos{left (x right )}$$
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.
$$a = -2$$
$$b = 5$$
$$c = 3$$
, то

D = b^2 – 4 * a * c =

(5)^2 – 4 * (-2) * (3) = 49

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = – frac{1}{2}$$
$$w_{2} = 3$$
делаем обратную замену
$$cos{left (x right )} = w$$
Дано уравнение
$$cos{left (x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
Или
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
, где n – любое целое число
подставляем w:
$$x_{1} = pi n + {acos}{left (w_{1} right )}$$
$$x_{1} = pi n + {acos}{left (- frac{1}{2} right )}$$
$$x_{1} = pi n + frac{2 pi}{3}$$
$$x_{2} = pi n + {acos}{left (w_{2} right )}$$
$$x_{2} = pi n + {acos}{left (3 right )}$$
$$x_{2} = pi n + {acos}{left (3 right )}$$
$$x_{3} = pi n + {acos}{left (w_{1} right )} – pi$$
$$x_{3} = pi n – pi + {acos}{left (- frac{1}{2} right )}$$
$$x_{3} = pi n – frac{pi}{3}$$
$$x_{4} = pi n + {acos}{left (w_{2} right )} – pi$$
$$x_{4} = pi n – pi + {acos}{left (3 right )}$$
$$x_{4} = pi n – pi + {acos}{left (3 right )}$$

-2*pi
x1 = —–
3

$$x_{1} = – frac{2 pi}{3}$$

2*pi
x2 = —-
3

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

/ ___
|/ 2 |
x3 = -2*I*atanh|—–|
2 /

$$x_{3} = – 2 i {atanh}{left (frac{sqrt{2}}{2} right )}$$

/ ___
|/ 2 |
x4 = 2*I*atanh|—–|
2 /

$$x_{4} = 2 i {atanh}{left (frac{sqrt{2}}{2} right )}$$

x1 = -67.0206432766000

x2 = -14.6607657168000

x3 = -20.9439510239000

x4 = -16.7551608191000

x5 = 29.3215314335000

x6 = 98.4365698125000

x7 = -10.4719755120000

x8 = 71.2094334814000

x9 = -8.37758040957000

x10 = 48.1710873550000

x11 = 58.6430628670000

x12 = 10.4719755120000

x13 = 20.9439510239000

x14 = -29.3215314335000

x15 = 27.2271363311000

x16 = -60.7374579694000

x17 = -104.719755120000

x18 = 14.6607657168000

x19 = 23.0383461263000

x20 = -41.8879020479000

x21 = -64.9262481742000

x22 = -71.2094334814000

x23 = 60.7374579694000

x24 = -52.3598775598000

x25 = -77.4926187885000

x26 = -39.7935069455000

x27 = 8.37758040957000

x28 = 92.1533845053000

x29 = 2.09439510239000

x30 = 35.6047167407000

x31 = -79.5870138909000

x32 = -85.8701991981000

x33 = 4.18879020479000

x34 = 67.0206432766000

x35 = 54.4542726622000

x36 = 46.0766922527000

x37 = -2.09439510239000

x38 = 41.8879020479000

x39 = -27.2271363311000

x40 = 90.0589894029000

x41 = -33.5103216383000

x42 = -83.7758040957000

x43 = 39.7935069455000

x44 = -73.3038285838000

x45 = -23.0383461263000

x46 = 77.4926187885000

x47 = -96.3421747101000

x48 = 73.3038285838000

x49 = -48.1710873550000

x50 = 64.9262481742000

x51 = -92.1533845053000

x52 = -46.0766922527000

x53 = -35.6047167407000

x54 = 79.5870138909000

x55 = 85.8701991981000

x56 = 96.3421747101000

x57 = -98.4365698125000

x58 = -178.023583703000

x59 = -54.4542726622000

x60 = 83.7758040957000

x61 = -586.430628670000

x62 = 16.7551608191000

x63 = 33.5103216383000

x64 = -90.0589894029000

x65 = -58.6430628670000

x66 = -159.174027782000

x67 = -4.18879020479000

x68 = 52.3598775598000