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

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

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

D = b^2 – 4 * a * c =

(3)^2 – 4 * (-1) * (-2) = 1

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

pi
x1 = —
2

$$x_{1} = frac{pi}{2}$$

/ / ___ / / ___
| |1 I*/ 3 || | |1 I*/ 3 ||
x2 = 2*re|atan|- – ——-|| + 2*I*im|atan|- – ——-||
2 2 // 2 2 //

$$x_{2} = 2 Re{left({atan}{left (frac{1}{2} – frac{sqrt{3} i}{2} right )}right)} + 2 i Im{left({atan}{left (frac{1}{2} – frac{sqrt{3} i}{2} right )}right)}$$

/ / ___ / / ___
| |1 I*/ 3 || | |1 I*/ 3 ||
x3 = 2*re|atan|- + ——-|| + 2*I*im|atan|- + ——-||
2 2 // 2 2 //

$$x_{3} = 2 Re{left({atan}{left (frac{1}{2} + frac{sqrt{3} i}{2} right )}right)} + 2 i Im{left({atan}{left (frac{1}{2} + frac{sqrt{3} i}{2} right )}right)}$$

x1 = -48.6946868672000

x2 = 95.8185747884000

x3 = 1.57079661902000

x4 = -98.9601678826000

x5 = 7.85398046563000

x6 = 83.2522046289000

x7 = 76.9690207794000

x8 = -67.5442432068000

x9 = 26.7035385470000

x10 = -92.6769828089000

x11 = -29.8451314931000

x12 = 51.8362789090000

x13 = 14.1371671182000

x14 = 1.57079700399000

x15 = 64.4026481915000

x16 = 14.1371682455000

x17 = 70.6858357115000

x18 = -4.71238848060000

x19 = -42.4115005431000

x20 = 39.2699074636000

x21 = -17.2787601164000

x22 = 39.2699086837000

x23 = 64.4026493045000

x24 = 95.8185760702000

x25 = 32.9867223887000

x26 = 45.5530925300000

x27 = 45.5530937812000

x28 = -67.5442408790000

x29 = 51.8362799898000

x30 = -86.3937989640000

x31 = -10.9955735517000

x32 = -92.6769845303000

x33 = -23.5619460759000

x34 = -23.5619450161000

x35 = -42.4115000881000

x36 = -325.154840065000

x37 = 7.85398285539000

x38 = 89.5353911753000

x39 = 89.5353909436000

x40 = 58.1194628122000

x41 = 45.5530940918000

x42 = -23.5619437178000

x43 = -92.6769840327000

x44 = -86.3937977050000

x45 = 20.4203510569000

x46 = -10.9955747753000

x47 = 20.4203521442000

x48 = -36.1283166952000

x49 = -98.9601691037000

x50 = -17.2787598788000

x51 = -29.8451289074000

x52 = -17.2787586177000

x53 = -36.1283154138000

x54 = -54.9778707172000

x55 = 70.6858344446000

x56 = 76.9690195526000

x57 = 1.57079536523000

x58 = -36.1283142806000

x59 = -67.5442421763000

x60 = -80.1106125755000

x61 = -73.8274272795000

x62 = 83.2522058482000

x63 = -42.4115017994000

x64 = -61.2610557825000

x65 = -73.8274260609000

x66 = -117.809725610000

x67 = 51.8362776268000

x68 = 89.5353896949000

x69 = 14.1371656592000

x70 = -86.3937971843000

x71 = -48.6946856448000

x72 = 20.4203534432000

x73 = 58.1194653977000

x74 = 102.101759966000

x75 = 1.57079785005000

x76 = -61.2610570408000

x77 = 64.4026506037000

x78 = 7.85398174771000

x79 = -80.1106114182000

x80 = 95.8185771224000

x81 = 58.1194643771000

x82 = 83.2522058457000

x83 = -54.9778719394000

x84 = -61.2610571936000

x85 = -4.71238970181000

x86 = -80.1106138557000

x87 = 70.6858340517000

x88 = 26.7035369654000

x89 = -29.8451300938000

x90 = 32.9867236139000

x91 = 26.7035372829000

x92 = -73.8274286446000