2*sin(x)^(2)+3*cos(x)-3=0

$$2 sin^{2}{left (x right )} + 3 cos{left (x right )} – 3 = 0$$

Дано уравнение
$$2 sin^{2}{left (x right )} + 3 cos{left (x right )} – 3 = 0$$
преобразуем
$$3 cos{left (x right )} – cos{left (2 x right )} – 2 = 0$$
$$- 2 cos^{2}{left (x right )} + 3 cos{left (x right )} – 1 = 0$$
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.
$$a = -2$$
$$b = 3$$
$$c = -1$$
, то

D = b^2 – 4 * a * c =

(3)^2 – 4 * (-2) * (-1) = 1

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

$$x_{1} = 0$$

-pi
x2 = —-
3

$$x_{2} = – frac{pi}{3}$$

pi
x3 = —
3

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

x1 = -89.0117918517000

x2 = -36.6519142919000

x3 = -50.2654823283000

x4 = 113.097336365000

x5 = 43.9822971693000

x6 = 42.9350995991000

x7 = 126.710903695000

x8 = -86.9173967493000

x9 = 50.2654824464000

x10 = 7.33038285838000

x11 = 63.8790506230000

x12 = 36.6519142919000

x13 = 26.1799387799000

x14 = 93.2005820565000

x15 = 49.2182849062000

x16 = 32.4631240871000

x17 = 68.0678408278000

x18 = 55.5014702134000

x19 = 12.5663704907000

x20 = -31.4159266584000

x21 = 95.2949771589000

x22 = 6.28318528439000

x23 = -19.8967534727000

x24 = -95.2949771589000

x25 = 87.9645943353000

x26 = 56.5486676413000

x27 = 17.8023583703000

x28 = 145.560459616000

x29 = -55.5014702134000

x30 = 1.04719755120000

x31 = -13.6135681656000

x32 = -87.9645943593000

x33 = 37.6991119735000

x34 = -13691.0607854000

x35 = -69.1150383715000

x36 = -61.7846555206000

x37 = -25.1327426220000

x38 = 13.6135681656000

x39 = -32.4631240871000

x40 = 57.5958653158000

x41 = -182.212372565000

x42 = 99.4837673637000

x43 = -1.04719755120000

x44 = 45.0294947015000

x45 = -93.2005820565000

x46 = 19.8967534727000

x47 = 0.0

x48 = 30.3687289847000

x49 = -17.8023583703000

x50 = -68.0678408278000

x51 = -57.5958653158000

x52 = -63.8790506230000

x53 = -45.0294947015000

x54 = 82.7286065445000

x55 = 100.530964793000

x56 = -80.6342114421000

x57 = 100.530964914000

x58 = -37.6991118768000

x59 = -43.9822971747000

x60 = -82.7286065445000

x61 = -12.5663692278000

x62 = -11.5191730632000

x63 = -94.2477794807000

x64 = -49.2182849062000

x65 = 51.3126800086000

x66 = -42.9350995991000

x67 = -24.0855436775000

x68 = -5.23598775598000

x69 = 70.1622359302000

x70 = 11.5191730632000

x71 = 94.2477796094000

x72 = 86.9173967493000

x73 = -30.3687289847000

x74 = 74.3510261350000

x75 = -38.7463093943000

x76 = 38.7463093943000

x77 = -51.3126800086000

x78 = -75.3982238055000

x79 = 61.7846555206000

x80 = -76.4454212374000

x81 = -81.6814090373000

x82 = -6.28318517680000

x83 = 62.8318525640000

x84 = -99.4837673637000

x85 = 81.6814091226000

x86 = 24.0855436775000

x87 = 5.23598775598000

x88 = 80.6342114421000

x89 = -7.33038285838000

x90 = 89.0117918517000

x91 = -74.3510261350000