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

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

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

D = b^2 – 4 * a * c =

(-1)^2 – 4 * (-2) * (0) = 1

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = – frac{1}{2}$$
$$w_{2} = 0$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (- frac{1}{2} right )}$$
$$x_{1} = 2 pi n – frac{pi}{6}$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (0 right )}$$
$$x_{2} = 2 pi n$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n – {asin}{left (- frac{1}{2} right )} + pi$$
$$x_{3} = 2 pi n + frac{7 pi}{6}$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n – {asin}{left (0 right )} + pi$$
$$x_{4} = 2 pi n + pi$$

$$x_{1} = 0$$

x2 = pi

$$x_{2} = pi$$

/ ___
| I / 3 |
x3 = -I*log|- – – —–|
2 2 /

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

/ ___
|/ 3 I|
x4 = -I*log|—– – -|
2 2/

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

x1 = 84.8230016469000

x2 = -12.5663706144000

x3 = 41.3643032723000

x4 = 78.5398163397000

x5 = 9.94837673637000

x6 = -2.61799387799000

x7 = -71.7330322570000

x8 = 47.6474885794000

x9 = 75.3982236862000

x10 = -75.3982236862000

x11 = 18.3259571459000

x12 = -97.3893722613000

x13 = 43.9822971503000

x14 = -34.0339204139000

x15 = -25.6563400043000

x16 = -75.9218224618000

x17 = -27.7507351067000

x18 = -3.14159265359000

x19 = -40.3171057211000

x20 = 62.3082542962000

x21 = 87.9645943005000

x22 = 44890.7410259000

x23 = -69.6386371546000

x24 = -84.8230016469000

x25 = 97.9129710369000

x26 = 12.0427718388000

x27 = -47.1238898038000

x28 = -31.9395253115000

x29 = -21.9911485751000

x30 = -37.6991118431000

x31 = 31.4159265359000

x32 = 21.9911485751000

x33 = -100.530964915000

x34 = 74.8746249106000

x35 = 47.1238898038000

x36 = 0.0

x37 = 16.2315620435000

x38 = -94.2477796077000

x39 = 100.007366139000

x40 = 50.2654824574000

x41 = -53.4070751110000

x42 = -59.6902604182000

x43 = 28.2743338823000

x44 = -43.9822971503000

x45 = -81.6814089933000

x46 = -90.5825881785000

x47 = 69.1150383790000

x48 = -46.6002910282000

x49 = -91.1061869541000

x50 = -63.3554518474000

x51 = -6.28318530718000

x52 = 68.5914396034000

x53 = -65.9734457254000

x54 = 72.2566310326000

x55 = 56.0250689890000

x56 = -56.5486677646000

x57 = 91.1061869541000

x58 = -62.8318530718000

x59 = -18.8495559215000

x60 = 94.2477796077000

x61 = 24.6091424531000

x62 = 91.6297857297000

x63 = -15.7079632679000

x64 = -88.4881930761000

x65 = 81.6814089933000

x66 = 34.5575191895000

x67 = -78.0162175641000

x68 = 65.9734457254000

x69 = 3.14159265359000

x70 = 40.8407044967000

x71 = 25.1327412287000

x72 = -34.5575191895000

x73 = -31.4159265359000

x74 = -84.2994028713000

x75 = -9.42477796077000

x76 = 6.28318530718000

x77 = 210.486707791000

x78 = -87.9645943005000

x79 = -50.2654824574000

x80 = 1538.85680148000

x81 = 53.9306738866000

x82 = 3.66519142919000

x83 = 37.6991118431000

x84 = 60.2138591938000

x85 = -19.3731546971000

x86 = -40.8407044967000