4*sin(x)^(2)=3

$$4 sin^{2}{left (x right )} = 3$$

Дано уравнение
$$4 sin^{2}{left (x right )} = 3$$
преобразуем
$$4 sin^{2}{left (x right )} – 3 = 0$$
$$4 sin^{2}{left (x right )} – 3 = 0$$
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.
$$a = 4$$
$$b = 0$$
$$c = -3$$
, то

D = b^2 – 4 * a * c =

(0)^2 – 4 * (4) * (-3) = 48

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

-pi
x1 = —-
3

$$x_{1} = – frac{pi}{3}$$

pi
x2 = —
3

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

2*pi
x3 = —-
3

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

4*pi
x4 = —-
3

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

x1 = -70.1622359302000

x2 = 11.5191730632000

x3 = -57.5958653158000

x4 = -14.6607657168000

x5 = 79.5870138909000

x6 = 98.4365698125000

x7 = 41.8879020479000

x8 = 83.7758040957000

x9 = 48.1710873550000

x10 = -13.6135681656000

x11 = -68.0678408278000

x12 = 10.4719755120000

x13 = 1515.29485658000

x14 = 20.9439510239000

x15 = -63.8790506230000

x16 = 38.7463093943000

x17 = 61.7846555206000

x18 = -79.5870138909000

x19 = -54.4542726622000

x20 = -83.7758040957000

x21 = 99.4837673637000

x22 = -61.7846555206000

x23 = -71.2094334814000

x24 = 60.7374579694000

x25 = -90.0589894029000

x26 = -77.4926187885000

x27 = -39.7935069455000

x28 = -41.8879020479000

x29 = 8.37758040957000

x30 = 92.1533845053000

x31 = 2.09439510239000

x32 = -378.038315982000

x33 = -32.4631240871000

x34 = 63.8790506230000

x35 = -85.8701991981000

x36 = 4.18879020479000

x37 = -60.7374579694000

x38 = 26.1799387799000

x39 = 54.4542726622000

x40 = 46.0766922527000

x41 = -2.09439510239000

x42 = 74.3510261350000

x43 = -17.8023583703000

x44 = -27.2271363311000

x45 = 90.0589894029000

x46 = -33.5103216383000

x47 = 76.4454212374000

x48 = -99.4837673637000

x49 = 39.7935069455000

x50 = 17.8023583703000

x51 = 32.4631240871000

x52 = 68.0678408278000

x53 = -96.3421747101000

x54 = 77.4926187885000

x55 = 55.5014702134000

x56 = -76.4454212374000

x57 = -48.1710873550000

x58 = -294.262511886000

x59 = -11.5191730632000

x60 = -46.0766922527000

x61 = -26.1799387799000

x62 = -35.6047167407000

x63 = 24.0855436775000

x64 = -49.2182849062000

x65 = 85.8701991981000

x66 = 96.3421747101000

x67 = -98.4365698125000

x68 = -24.0855436775000

x69 = -30.3687289847000

x70 = -19.8967534727000

x71 = 16.7551608191000

x72 = -93.2005820565000

x73 = 33.5103216383000

x74 = -5.23598775598000

x75 = 19.8967534727000

x76 = 82.7286065445000

x77 = -10.4719755120000

x78 = -92.1533845053000

x79 = -4.18879020479000

x80 = 52.3598775598000

x81 = -55.5014702134000

x82 = 70.1622359302000

x83 = 30.3687289847000