cos(2*x)+sqrt(2)*sin(x)+1=0

$$sqrt{2} sin{left (x right )} + cos{left (2 x right )} + 1 = 0$$

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

D = b^2 – 4 * a * c =

(sqrt(2))^2 – 4 * (-2) * (2) = 18

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = – frac{sqrt{2}}{2}$$
$$w_{2} = sqrt{2}$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (- frac{sqrt{2}}{2} right )}$$
$$x_{1} = 2 pi n – frac{pi}{4}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (sqrt{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (sqrt{2} right )}$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n – {asin}{left (- frac{sqrt{2}}{2} right )} + pi$$
$$x_{3} = 2 pi n + frac{5 pi}{4}$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n + pi – {asin}{left (sqrt{2} right )}$$
$$x_{4} = 2 pi n + pi – {asin}{left (sqrt{2} right )}$$

3*pi /log(2) / ___
x1 = – —- + I*|—— – log/ 2 /|
4 2 /

$$x_{1} = – frac{3 pi}{4} + i left(- log{left (sqrt{2} right )} + frac{1}{2} log{left (2 right )}right)$$

pi /log(2) / ___
x2 = – — + I*|—— – log/ 2 /|
4 2 /

$$x_{2} = – frac{pi}{4} + i left(- log{left (sqrt{2} right )} + frac{1}{2} log{left (2 right )}right)$$

pi /| ___|
x3 = — – I*log|I + I*/ 2 |/
2

$$x_{3} = frac{pi}{2} – i log{left (left|{i + sqrt{2} i}right| right )}$$

pi / ___
x4 = — – I*log -1 + / 2 /
2

$$x_{4} = frac{pi}{2} – i log{left (-1 + sqrt{2} right )}$$

x1 = -77.7544181763000

x2 = -33.7721210261000

x3 = 10.2101761242000

x4 = 85.6083998103000

x5 = 22.7765467385000

x6 = 62.0464549084000

x7 = 74.6128255228000

x8 = -63.6172512352000

x9 = -76.1836218496000

x10 = 18.0641577581000

x11 = 66.7588438888000

x12 = 68.3296402156000

x13 = -13.3517687778000

x14 = -71.4712328692000

x15 = -82.4668071567000

x16 = -2.35619449019000

x17 = -69.9004365424000

x18 = 3.92699081699000

x19 = 60.4756585816000

x20 = 24.3473430653000

x21 = 93.4623814443000

x22 = -57.3340659280000

x23 = -38.4845100065000

x24 = -46.3384916404000

x25 = 49.4800842940000

x26 = 99.7455667515000

x27 = 47.9092879672000

x28 = 16.4933614313000

x29 = 11.7809724510000

x30 = -574.126057444000

x31 = 98.1747704247000

x32 = 41.6261026601000

x33 = -19.6349540849000

x34 = -32.2013246993000

x35 = 5.49778714378000

x36 = -21.2057504117000

x37 = 30.6305283725000

x38 = -40.0553063333000

x39 = 87.1791961371000

x40 = -84.0376034835000

x41 = -27.4889357189000

x42 = -90.3207887907000

x43 = -25.9181393921000

x44 = 55.7632696012000

x45 = 91.8915851175000

x46 = 54.1924732744000

x47 = -65.1880475620000