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

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

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

D = b^2 – 4 * a * c =

(-1)^2 – 4 * (5) * (0) = 1

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = frac{1}{5}$$
$$w_{2} = 0$$
делаем обратную замену
$$cos{left (2 x right )} = w$$
Дано уравнение
$$cos{left (2 x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$2 x = pi n + {acos}{left (w right )}$$
$$2 x = pi n + {acos}{left (w right )} – pi$$
Или
$$2 x = pi n + {acos}{left (w right )}$$
$$2 x = pi n + {acos}{left (w right )} – pi$$
, где n – любое целое число
Разделим обе части полученного ур-ния на
$$2$$
подставляем w:
$$x_{1} = frac{pi n}{2} + frac{1}{2} {acos}{left (w_{1} right )}$$
$$x_{1} = frac{pi n}{2} + frac{1}{2} {acos}{left (frac{1}{5} right )}$$
$$x_{1} = frac{pi n}{2} + frac{1}{2} {acos}{left (frac{1}{5} right )}$$
$$x_{2} = frac{pi n}{2} + frac{1}{2} {acos}{left (w_{2} right )}$$
$$x_{2} = frac{pi n}{2} + frac{1}{2} {acos}{left (0 right )}$$
$$x_{2} = frac{pi n}{2} + frac{pi}{4}$$
$$x_{3} = frac{pi n}{2} + frac{1}{2} {acos}{left (w_{1} right )} – frac{pi}{2}$$
$$x_{3} = frac{pi n}{2} – frac{pi}{2} + frac{1}{2} {acos}{left (frac{1}{5} right )}$$
$$x_{3} = frac{pi n}{2} – frac{pi}{2} + frac{1}{2} {acos}{left (frac{1}{5} right )}$$
$$x_{4} = frac{pi n}{2} + frac{1}{2} {acos}{left (w_{2} right )} – frac{pi}{2}$$
$$x_{4} = frac{pi n}{2} – frac{pi}{2} + frac{1}{2} {acos}{left (0 right )}$$
$$x_{4} = frac{pi n}{2} – frac{pi}{4}$$

pi
x1 = —
4

$$x_{1} = frac{pi}{4}$$

3*pi
x2 = —-
4

$$x_{2} = frac{3 pi}{4}$$

acos(1/5)
x3 = pi – ———
2

$$x_{3} = – frac{1}{2} {acos}{left (frac{1}{5} right )} + pi$$

acos(1/5)
x4 = ———
2

$$x_{4} = frac{1}{2} {acos}{left (frac{1}{5} right )}$$

x1 = -77.7544181763000

x2 = 90.3207887907000

x3 = -49.5807632544000

x4 = 22.7765467385000

x5 = -13.3517687778000

x6 = 66.7588438888000

x7 = 69.9004365424000

x8 = -22.7765467385000

x9 = 63.6172512352000

x10 = -10.1094971638000

x11 = 30.6305283725000

x12 = 54.1924732744000

x13 = 2.35619449019000

x14 = 38.3838310461000

x15 = -98.0740914643000

x16 = 10.2101761242000

x17 = -98.1747704247000

x18 = -54.0917943140000

x19 = 76.1836218496000

x20 = -2.35619449019000

x21 = -93.5630604047000

x22 = -55.7632696012000

x23 = 99.8462457119000

x24 = -38.4845100065000

x25 = -46.3384916404000

x26 = 49.4800842940000

x27 = -47.8086090068000

x28 = 41.6261026601000

x29 = -79.3252145031000

x30 = -18.0641577581000

x31 = -62.0464549084000

x32 = 1416.07288861000

x33 = 44.7676953137000

x34 = 46.3384916404000

x35 = -13794.0486227000

x36 = -11.7809724510000

x37 = 27.4889357189000

x38 = 16.3926824710000

x39 = 85.6083998103000

x40 = -902.422489744000

x41 = 32.2013246993000

x42 = 74.6128255228000

x43 = -63.6172512352000

x44 = -33.7721210261000

x45 = -25.8174604317000

x46 = -99.7455667515000

x47 = -60.4756585816000

x48 = -90.3207887907000

x49 = -16.4933614313000

x50 = 71.4712328692000

x51 = 88.7499924639000

x52 = 3.92699081699000

x53 = 93.4623814443000

x54 = 60.3749796212000

x55 = -91.7909061571000

x56 = -76.0829428892000

x57 = -19.6349540849000

x58 = 11.8816514114000

x59 = 77.8550971367000

x60 = 24.3473430653000

x61 = -71.5719118296000

x62 = -84.0376034835000

x63 = -35.3429173529000

x64 = -41.6261026601000

x65 = 82.3661281963000

x66 = 0.785398163397000

x67 = 96.6039740979000

x68 = 25.9181393921000

x69 = -24.3473430653000

x70 = 55.8639485616000

x71 = 6221.13885227000

x72 = -40.0553063333000

x73 = -0.785398163397000

x74 = -85.6083998103000

x75 = -57.3340659280000

x76 = -82.3661281963000

x77 = 40.1559852937000

x78 = -5.59846610418000

x79 = -32.1006457389000

x80 = 47.9092879672000

x81 = -27.5896146793000

x82 = 18.1648367185000

x83 = 68.3296402156000

x84 = 19.6349540849000

x85 = 5.49778714378000

x86 = 33.8727999865000

x87 = 52.6216769476000

x88 = -68.3296402156000

x89 = -69.7997575820000

x90 = 91.8915851175000

x91 = 8.63937979737000