cos(x)^(2)=(1/2)

$$cos^{2}{left (x right )} = frac{1}{2}$$

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

D = b^2 – 4 * a * c =

(0)^2 – 4 * (1) * (-1/2) = 2

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

pi
x1 = —
4

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

3*pi
x2 = —-
4

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

5*pi
x3 = —-
4

$$x_{3} = frac{5 pi}{4}$$

7*pi
x4 = —-
4

$$x_{4} = frac{7 pi}{4}$$

x1 = -77.7544181763000

x2 = 14247.9080822000

x3 = 90.3207887907000

x4 = 22.7765467385000

x5 = -93.4623814443000

x6 = 77.7544181763000

x7 = -13.3517687778000

x8 = -71.4712328692000

x9 = 33.7721210261000

x10 = -47.9092879672000

x11 = 66.7588438888000

x12 = 162.577419823000

x13 = 69.9004365424000

x14 = -25.9181393921000

x15 = 63.6172512352000

x16 = 30.6305283725000

x17 = -49.4800842940000

x18 = 84.0376034835000

x19 = 54.1924732744000

x20 = 2.35619449019000

x21 = -1131.75875346000

x22 = -33.7721210261000

x23 = 10.2101761242000

x24 = 87.1791961371000

x25 = 76.1836218496000

x26 = 49.4800842940000

x27 = -2.35619449019000

x28 = -5.49778714378000

x29 = -55.7632696012000

x30 = 60.4756585816000

x31 = -54.1924732744000

x32 = -38.4845100065000

x33 = -46.3384916404000

x34 = 40.0553063333000

x35 = 41.6261026601000

x36 = -32.2013246993000

x37 = -79.3252145031000

x38 = -18.0641577581000

x39 = -62.0464549084000

x40 = 44.7676953137000

x41 = 46.3384916404000

x42 = -11.7809724510000

x43 = 27.4889357189000

x44 = 85.6083998103000

x45 = 32.2013246993000

x46 = 74.6128255228000

x47 = -63.6172512352000

x48 = -76.1836218496000

x49 = 18.0641577581000

x50 = -99.7455667515000

x51 = -60.4756585816000

x52 = -90.3207887907000

x53 = -16.4933614313000

x54 = -69.9004365424000

x55 = 88.7499924639000

x56 = 3.92699081699000

x57 = 11.7809724510000

x58 = 98.1747704247000

x59 = -19.6349540849000

x60 = 38.4845100065000

x61 = 24.3473430653000

x62 = 62.0464549084000

x63 = -84.0376034835000

x64 = -35.3429173529000

x65 = -41.6261026601000

x66 = -91.8915851175000

x67 = 82.4668071567000

x68 = 96.6039740979000

x69 = 25.9181393921000

x70 = -27.4889357189000

x71 = 384.059701901000

x72 = -82.4668071567000

x73 = -10.2101761242000

x74 = -40.0553063333000

x75 = -85.6083998103000

x76 = -57.3340659280000

x77 = -98.1747704247000

x78 = 47.9092879672000

x79 = 16.4933614313000

x80 = -12461.9126586000

x81 = -3.92699081699000

x82 = 68.3296402156000

x83 = 19.6349540849000

x84 = 5.49778714378000

x85 = 99.7455667515000

x86 = 52.6216769476000

x87 = -24.3473430653000

x88 = -68.3296402156000

x89 = 55.7632696012000

x90 = 91.8915851175000

x91 = 8.63937979737000