5*cos(2*x)+4*cos(x)-1=0

$$4 cos{left (x right )} + 5 cos{left (2 x right )} – 1 = 0$$

Дано уравнение
$$4 cos{left (x right )} + 5 cos{left (2 x right )} – 1 = 0$$
преобразуем
$$10 cos^{2}{left (x right )} + 4 cos{left (x right )} – 6 = 0$$
$$10 cos^{2}{left (x right )} + 4 cos{left (x right )} – 6 = 0$$
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.
$$a = 10$$
$$b = 4$$
$$c = -6$$
, то

D = b^2 – 4 * a * c =

(4)^2 – 4 * (10) * (-6) = 256

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

$$x_{1} = pi$$

/3 4*I
x2 = -I*log|- – —|
5 5 /

$$x_{2} = – i log{left (frac{3}{5} – frac{4 i}{5} right )}$$

/3 4*I
x3 = -I*log|- + —|
5 5 /

$$x_{3} = – i log{left (frac{3}{5} + frac{4 i}{5} right )}$$

x1 = 74.4709284682000

x2 = -99.6036696969000

x3 = 3.14159259002000

x4 = -68.1877431610000

x5 = -82.6087042113000

x6 = -26.0600364467000

x7 = -15.7079632965000

x8 = 78.5398161898000

x9 = -40.8407045201000

x10 = 17.9222607035000

x11 = -17.9222607035000

x12 = 13.4936658324000

x13 = -72.2566308763000

x14 = -11.6390753964000

x15 = 9.42477819298000

x16 = -93.3204843897000

x17 = 91.1061868320000

x18 = 68.1877431610000

x19 = -78.5398161101000

x20 = -40.8407045022000

x21 = 40.8407042720000

x22 = 61.9045578538000

x23 = -63.7591482898000

x24 = -49.3381872394000

x25 = -5.35589008918000

x26 = 308.803375270000

x27 = -70.0423335970000

x28 = 38.6264070611000

x29 = -57.4759629826000

x30 = 57.4759629826000

x31 = 84.8230014236000

x32 = 91.1061870131000

x33 = 11.6390753964000

x34 = -38.6264070611000

x35 = 99.6036696969000

x36 = -91.1061871795000

x37 = -24.2054460107000

x38 = 59.6902605943000

x39 = -97.3893724368000

x40 = 36.7718166251000

x41 = 15.7079634377000

x42 = -76.3255189042000

x43 = 40.8407045618000

x44 = 72.2566310277000

x45 = 63.7591482898000

x46 = -84.8230014876000

x47 = -55.6213725466000

x48 = -30.4886313179000

x49 = -32.3432217539000

x50 = 82.6087042113000

x51 = 47.1238898961000

x52 = -53.4070752806000

x53 = 26.0600364467000

x54 = 32.3432217539000

x55 = 80.7541137753000

x56 = 126.591001362000

x57 = -21.9911485865000

x58 = 47.1238897080000

x59 = 65.9734457529000

x60 = -13.4936658324000

x61 = -61.9045578538000

x62 = 24.2054460107000

x63 = 3.14159277279000

x64 = -47.1238900296000

x65 = 70.0423335970000

x66 = -9.42477812402000

x67 = -34.5575189583000

x68 = 34.5575190328000

x69 = -28.2743337192000

x70 = -65.9734457650000

x71 = -3.14159287911000

x72 = 30.4886313179000

x73 = 19.7768511395000

x74 = -59.6902604576000

x75 = 76.3255189042000

x76 = -40.8407043554000

x77 = -84.8230016204000

x78 = 47.1238897529000

x79 = -74.4709284682000

x80 = 21.9911485852000

x81 = 28.2743338652000

x82 = -19.7768511395000

x83 = -51.1927776754000

x84 = 55.6213725466000

x85 = 97.3893724940000

x86 = 53.4070753438000

x87 = -7.21048052518000