5+cos(x)=6*sin(x)^(2)

$$cos{left (x right )} + 5 = 6 sin^{2}{left (x right )}$$

Дано уравнение
$$cos{left (x right )} + 5 = 6 sin^{2}{left (x right )}$$
преобразуем
$$6 cos^{2}{left (x right )} + cos{left (x right )} – 1 = 0$$
$$6 cos^{2}{left (x right )} + cos{left (x right )} – 1 = 0$$
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.
$$a = 6$$
$$b = 1$$
$$c = -1$$
, то

D = b^2 – 4 * a * c =

(1)^2 – 4 * (6) * (-1) = 25

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

-2*pi
x1 = —–
3

$$x_{1} = – frac{2 pi}{3}$$

2*pi
x2 = —-
3

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

/ ___
|/ 2 |
x3 = -2*atan|—–|
2 /

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

/ ___
|/ 2 |
x4 = 2*atan|—–|
2 /

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

x1 = -93.0168201904000

x2 = 57.7796271820000

x3 = 79.5870138909000

x4 = 48.1710873550000

x5 = 52.3598775598000

x6 = -30.1849671186000

x7 = 26.3637006461000

x8 = -71.2094334814000

x9 = -64.0628124891000

x10 = 92.1533845053000

x11 = 67.8840789616000

x12 = -26.3637006461000

x13 = 80.4504495760000

x14 = -17.6185965042000

x15 = -2.09439510239000

x16 = -33.5103216383000

x17 = 39.7935069455000

x18 = -99.3000054975000

x19 = 77.4926187885000

x20 = -92.1533845053000

x21 = 4.18879020479000

x22 = -23.9017818114000

x23 = -90.0589894029000

x24 = -70.3459977963000

x25 = -4.18879020479000

x26 = 41.8879020479000

x27 = -5.05222588984000

x28 = 32.6468859532000

x29 = 70.3459977963000

x30 = -77.4926187885000

x31 = 8.37758040957000

x32 = 74.1672642688000

x33 = 54.4542726622000

x34 = -51.4964418748000

x35 = -27.2271363311000

x36 = -340.522966005000

x37 = -23.0383461263000

x38 = 17.6185965042000

x39 = -46.0766922527000

x40 = 30.1849671186000

x41 = 83.7758040957000

x42 = 76.6291831035000

x43 = 55.3177083473000

x44 = -82.9123684107000

x45 = 20.0805153389000

x46 = -13.7973300317000

x47 = -20.0805153389000

x48 = 13.7973300317000

x49 = -10.4719755120000

x50 = 99.3000054975000

x51 = 237.530082255000

x52 = -41.8879020479000

x53 = -57.7796271820000

x54 = 60.7374579694000

x55 = 46.0766922527000

x56 = 90.0589894029000

x57 = 85.8701991981000

x58 = 96.3421747101000

x59 = -98.4365698125000

x60 = 36.4681524257000

x61 = 33.5103216383000

x62 = -49.0345230401000

x63 = -1.23095941734000

x64 = 98.4365698125000

x65 = -55.3177083473000

x66 = 10.4719755120000

x67 = 38.9300712604000

x68 = -79.5870138909000

x69 = -11.3354111970000

x70 = -83.7758040957000

x71 = -7.51414472452000

x72 = 61.6008936545000

x73 = -74.1672642688000

x74 = -39.7935069455000

x75 = -32.6468859532000

x76 = 2.09439510239000

x77 = -85.8701991981000

x78 = -61.6008936545000

x79 = 64.0628124891000

x80 = -67.8840789616000

x81 = 82.9123684107000

x82 = 11.3354111970000

x83 = -48.1710873550000

x84 = 23.9017818114000

x85 = -35.6047167407000

x86 = -54.4542726622000

x87 = 16.7551608191000

x88 = -76.6291831035000

x89 = -506.843614779000