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
$$sin{left (x right )} + cos{left (2 x right )} – 1 = 0$$
$$- 2 sin^{2}{left (x right )} + sin{left (x right )} = 0$$
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
$$w_{1} = 0$$
$$w_{2} = frac{1}{2}$$
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:
$$x_{1} = 2 pi n + {asin}{left (w_{1} right )}$$
$$x_{1} = 2 pi n + {asin}{left (0 right )}$$
$$x_{1} = 2 pi n$$
$$x_{2} = 2 pi n + {asin}{left (w_{2} right )}$$
$$x_{2} = 2 pi n + {asin}{left (frac{1}{2} right )}$$
$$x_{2} = 2 pi n + frac{pi}{6}$$
$$x_{3} = 2 pi n – {asin}{left (w_{1} right )} + pi$$
$$x_{3} = 2 pi n – {asin}{left (0 right )} + pi$$
$$x_{3} = 2 pi n + pi$$
$$x_{4} = 2 pi n – {asin}{left (w_{2} right )} + pi$$
$$x_{4} = 2 pi n – {asin}{left (frac{1}{2} right )} + pi$$
$$x_{4} = 2 pi n + frac{5 pi}{6}$$

$$x_{1} = 0$$

x2 = pi

$$x_{2} = pi$$

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

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

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

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

x1 = -100.007366139000

x2 = 50.7890812330000

x3 = -34.5575191895000

x4 = 97.3893722613000

x5 = 9.42477796077000

x6 = 56.5486677646000

x7 = -24.6091424531000

x8 = -12.0427718388000

x9 = 128.281700022000

x10 = 18.8495559215000

x11 = -41.3643032723000

x12 = 31.9395253115000

x13 = 25.6563400043000

x14 = -5.75958653158000

x15 = 12.5663706144000

x16 = -47.6474885794000

x17 = 19.3731546971000

x18 = -78.5398163397000

x19 = 43.9822971503000

x20 = 50.2654824574000

x21 = -62.3082542962000

x22 = -49.7418836818000

x23 = -59.6902604182000

x24 = 62.8318530718000

x25 = 87.9645943005000

x26 = 46.6002910282000

x27 = -106.290551446000

x28 = -68.5914396034000

x29 = -84.8230016469000

x30 = 75.9218224618000

x31 = 59.6902604182000

x32 = -97.9129710369000

x33 = -21.9911485751000

x34 = -37.6991118431000

x35 = 78.0162175641000

x36 = 21.9911485751000

x37 = 47.1238898038000

x38 = 0.0

x39 = -476.998484570000

x40 = -367.566340470000

x41 = 2.61799387799000

x42 = 84.2994028713000

x43 = 15.7079632679000

x44 = 69.6386371546000

x45 = -53.4070751110000

x46 = -93.7241808321000

x47 = 28.2743338823000

x48 = -25.1327412287000

x49 = -43.9822971503000

x50 = -81.6814089933000

x51 = 69.1150383790000

x52 = -85.3466004225000

x53 = 72.2566310326000

x54 = -74.8746249106000

x55 = -65.9734457254000

x56 = -72.2566310326000

x57 = -56.0250689890000

x58 = -18.3259571459000

x59 = -3.66519142919000

x60 = 34.0339204139000

x61 = 91.1061869541000

x62 = -62.8318530718000

x63 = -18.8495559215000

x64 = 94.2477796077000

x65 = -9.94837673637000

x66 = 82.2050077689000

x67 = -15.7079632679000

x68 = 65.9734457254000

x69 = 3.14159265359000

x70 = 100.530964915000

x71 = 25.1327412287000

x72 = 38.2227106187000

x73 = 40.3171057211000

x74 = -31.4159265359000

x75 = -75.3982236862000

x76 = -69.1150383790000

x77 = -53.9306738866000

x78 = 6.28318530718000

x79 = -87.9645943005000

x80 = 90.5825881785000

x81 = -91.6297857297000

x82 = 53.4070751110000

x83 = 96.8657734857000

x84 = -40.8407044967000

x85 = -143.989663290000

x86 = -28.2743338823000