4*sin(x)^(2)/2-1=0

$$frac{4}{2} sin^{2}{left (x right )} – 1 = 0$$

Дано уравнение
$$frac{4}{2} sin^{2}{left (x right )} – 1 = 0$$
преобразуем
$$- cos{left (2 x right )} = 0$$
$$frac{4}{2} sin^{2}{left (x right )} – 1 = 0$$
Сделаем замену
$$w = sin{left (x right )}$$
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.
$$a = 2$$
$$b = 0$$
$$c = -1$$
, то

D = b^2 – 4 * a * c =

(0)^2 – 4 * (2) * (-1) = 8

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
$$sin{left (x right )} = w$$
Дано уравнение
$$sin{left (x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 pi n + {asin}{left (w right )}$$
$$x = 2 pi n – {asin}{left (w right )} + pi$$
Или
$$x = 2 pi n + {asin}{left (w right )}$$
$$x = 2 pi n – {asin}{left (w right )} + pi$$
, где n – любое целое число
подставляем w:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (frac{sqrt{2}}{2} right )}$$
$$x_{1} = 2 pi n + frac{pi}{4}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (- frac{sqrt{2}}{2} right )}$$
$$x_{2} = 2 pi n – frac{pi}{4}$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n – {asin}{left (frac{sqrt{2}}{2} right )} + pi$$
$$x_{3} = 2 pi n + frac{3 pi}{4}$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n – {asin}{left (- frac{sqrt{2}}{2} right )} + pi$$
$$x_{4} = 2 pi n + frac{5 pi}{4}$$

-pi
x1 = —-
4

$$x_{1} = – frac{pi}{4}$$

pi
x2 = —
4

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

3*pi
x3 = —-
4

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

5*pi
x4 = —-
4

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

x1 = -77.7544181763000

x2 = 90.3207887907000

x3 = 22.7765467385000

x4 = -93.4623814443000

x5 = 77.7544181763000

x6 = -13.3517687778000

x7 = -71.4712328692000

x8 = 33.7721210261000

x9 = -47.9092879672000

x10 = 66.7588438888000

x11 = 162.577419823000

x12 = 69.9004365424000

x13 = -25.9181393921000

x14 = 63.6172512352000

x15 = 30.6305283725000

x16 = -49.4800842940000

x17 = 84.0376034835000

x18 = 54.1924732744000

x19 = 2.35619449019000

x20 = -33.7721210261000

x21 = 10.2101761242000

x22 = 87.1791961371000

x23 = 76.1836218496000

x24 = 49.4800842940000

x25 = -2.35619449019000

x26 = -5.49778714378000

x27 = -55.7632696012000

x28 = 60.4756585816000

x29 = -54.1924732744000

x30 = -38.4845100065000

x31 = -46.3384916404000

x32 = 40.0553063333000

x33 = 41.6261026601000

x34 = -32.2013246993000

x35 = -79.3252145031000

x36 = -18.0641577581000

x37 = -62.0464549084000

x38 = 44.7676953137000

x39 = 46.3384916404000

x40 = -11.7809724510000

x41 = 27.4889357189000

x42 = 85.6083998103000

x43 = 32.2013246993000

x44 = 74.6128255228000

x45 = -63.6172512352000

x46 = -76.1836218496000

x47 = 18.0641577581000

x48 = -99.7455667515000

x49 = -60.4756585816000

x50 = -90.3207887907000

x51 = -16.4933614313000

x52 = -69.9004365424000

x53 = 88.7499924639000

x54 = 3.92699081699000

x55 = 11.7809724510000

x56 = 98.1747704247000

x57 = -19.6349540849000

x58 = 38.4845100065000

x59 = 24.3473430653000

x60 = 62.0464549084000

x61 = -84.0376034835000

x62 = -35.3429173529000

x63 = -41.6261026601000

x64 = -91.8915851175000

x65 = 82.4668071567000

x66 = 96.6039740979000

x67 = -1144.32512407000

x68 = 25.9181393921000

x69 = -27.4889357189000

x70 = 384.059701901000

x71 = -82.4668071567000

x72 = -10.2101761242000

x73 = -40.0553063333000

x74 = -85.6083998103000

x75 = -57.3340659280000

x76 = -98.1747704247000

x77 = 47.9092879672000

x78 = 16.4933614313000

x79 = -3.92699081699000

x80 = 68.3296402156000

x81 = 19.6349540849000

x82 = 5.49778714378000

x83 = 99.7455667515000

x84 = 52.6216769476000

x85 = -24.3473430653000

x86 = -68.3296402156000

x87 = 55.7632696012000

x88 = 91.8915851175000

x89 = 8.63937979737000