6*cos(x)^(2)+5*cos(x)-6=0

$$6 cos^{2}{left (x right )} + 5 cos{left (x right )} – 6 = 0$$

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

D = b^2 – 4 * a * c =

(5)^2 – 4 * (6) * (-6) = 169

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

$$x_{1} = – {acos}{left (frac{2}{3} right )} + 2 pi$$

x2 = acos(2/3)

$$x_{2} = {acos}{left (frac{2}{3} right )}$$

x3 = -re(acos(-3/2)) + 2*pi – I*im(acos(-3/2))

$$x_{3} = – Re{left({acos}{left (- frac{3}{2} right )}right)} + 2 pi – i Im{left({acos}{left (- frac{3}{2} right )}right)}$$

x4 = I*im(acos(-3/2)) + re(acos(-3/2))

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

x1 = 43.1412284797000

x2 = 44.8233658208000

x3 = 32.2569952065000

x4 = -11.7253019438000

x5 = -0.841068670568000

x6 = -95.0888482783000

x7 = -87.1235256299000

x8 = 7.12425397775000

x9 = -76.2392923567000

x10 = 57.3897364352000

x11 = -99.6898962443000

x12 = -49.4244137869000

x13 = 93.4067109371000

x14 = 49.4244137869000

x15 = 68.2739697084000

x16 = 55.7075990940000

x17 = 5.44211663661000

x18 = -61.9907844012000

x19 = -69.9561070495000

x20 = 95.0888482783000

x21 = 18.0084872510000

x22 = -51.1065511280000

x23 = -32.2569952065000

x24 = 63.6729217424000

x25 = -57.3897364352000

x26 = 24.2916725582000

x27 = 88.8056629711000

x28 = -63.6729217424000

x29 = -30.5748578653000

x30 = 131.105822780000

x31 = 19.6906245921000

x32 = 87.1235256299000

x33 = 358.982631180000

x34 = -55.7075990940000

x35 = 80.8403403228000

x36 = 69.9561070495000

x37 = -93.4067109371000

x38 = -82.5224776639000

x39 = 13.4074392849000

x40 = 99.6898962443000

x41 = 25.9738098993000

x42 = 658.893388583000

x43 = -18.0084872510000

x44 = -68.2739697084000

x45 = -6332.60972097000

x46 = 82.5224776639000

x47 = 36.8580431725000

x48 = 0.841068670568000

x49 = 74.5571550156000

x50 = 38.5401805136000

x51 = -24.2916725582000

x52 = -13.4074392849000

x53 = 11.7253019438000

x54 = 61.9907844012000

x55 = -38.5401805136000

x56 = -74.5571550156000

x57 = -5.44211663661000

x58 = 30.5748578653000

x59 = 76.2392923567000

x60 = -19.6906245921000

x61 = -7.12425397775000

x62 = -25.9738098993000