1+2*sin(x)-cos(x)^(2)=0

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

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

D = b^2 – 4 * a * c =

(2)^2 – 4 * (1) * (0) = 4

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = 0$$
$$w_{2} = -2$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (0 right )}$$
$$x_{1} = 2 pi n$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (-2 right )}$$
$$x_{2} = 2 pi n – {asin}{left (2 right )}$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n – {asin}{left (0 right )} + pi$$
$$x_{3} = 2 pi n + pi$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n + pi – {asin}{left (-2 right )}$$
$$x_{4} = 2 pi n + pi + {asin}{left (2 right )}$$

$$x_{1} = 0$$

/ / ___ / / ___
| |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 = -94.2477796077000

x2 = 31.4159265359000

x3 = 81.6814089933000

x4 = 84.8230016469000

x5 = -53.4070751110000

x6 = 65.9734457254000

x7 = 3.14159265359000

x8 = 15.7079632679000

x9 = 100.530964915000

x10 = 50.2654824574000

x11 = -3.14159265359000

x12 = 40.8407044967000

x13 = -59.6902604182000

x14 = 97.3893722613000

x15 = 78.5398163397000

x16 = -25.1327412287000

x17 = -43.9822971503000

x18 = 25.1327412287000

x19 = 62.8318530718000

x20 = -91.1061869541000

x21 = 87.9645943005000

x22 = 69.1150383790000

x23 = -34.5575191895000

x24 = 28.2743338823000

x25 = -31.4159265359000

x26 = 37.6991118431000

x27 = -28.2743338823000

x28 = 72.2566310326000

x29 = 56.5486677646000

x30 = -75.3982236862000

x31 = -69.1150383790000

x32 = -6.28318530718000

x33 = -9.42477796077000

x34 = 6.28318530718000

x35 = 75.3982236862000

x36 = -65.9734457254000

x37 = -87.9645943005000

x38 = -72.2566310326000

x39 = 18.8495559215000

x40 = -84.8230016469000

x41 = 9.42477796077000

x42 = -50.2654824574000

x43 = -56.5486677646000

x44 = 43.9822971503000

x45 = 91.1061869541000

x46 = 59.6902604182000

x47 = -47.1238898038000

x48 = 12.5663706144000

x49 = -414.690230274000

x50 = -81.6814089933000

x51 = -18.8495559215000

x52 = -12.5663706144000

x53 = -37.6991118431000

x54 = -97.3893722613000

x55 = 94.2477796077000

x56 = 34.5575191895000

x57 = -1803.27418316000

x58 = -21.9911485751000

x59 = 21.9911485751000

x60 = -100.530964915000

x61 = 53.4070751110000

x62 = -113.097335529000

x63 = -78.5398163397000

x64 = 0.0

x65 = -62.8318530718000

x66 = -40.8407044967000

x67 = -15.7079632679000

x68 = 47.1238898038000