cos(2*x)+3*sin(x)=1

$$3 sin{left (x right )} + cos{left (2 x right )} = 1$$

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

D = b^2 – 4 * a * c =

(3)^2 – 4 * (-2) * (0) = 9

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

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

w2 = (-b – sqrt(D)) / (2*a)

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

$$x_{1} = 0$$

x2 = pi

$$x_{2} = pi$$

/| ___|
pi ||3*I I*/ 5 ||
x3 = — – I*log||— – ——-||
2 | 2 2 |/

$$x_{3} = frac{pi}{2} – i log{left (left|{- frac{sqrt{5} i}{2} + frac{3 i}{2}}right| right )}$$

/| ___|
pi ||3*I I*/ 5 ||
x4 = — – I*log||— + ——-||
2 | 2 2 |/

$$x_{4} = frac{pi}{2} – i log{left (left|{frac{sqrt{5} i}{2} + frac{3 i}{2}}right| right )}$$

x1 = -94.2477796077000

x2 = -109.955742876000

x3 = 81.6814089933000

x4 = 84.8230016469000

x5 = -21.9911485751000

x6 = -53.4070751110000

x7 = -631.460123372000

x8 = 65.9734457254000

x9 = 3.14159265359000

x10 = 15.7079632679000

x11 = 100.530964915000

x12 = 50.2654824574000

x13 = 207.345115137000

x14 = -59.6902604182000

x15 = 97.3893722613000

x16 = 78.5398163397000

x17 = -25.1327412287000

x18 = -43.9822971503000

x19 = 25.1327412287000

x20 = -81.6814089933000

x21 = -91.1061869541000

x22 = 87.9645943005000

x23 = 69.1150383790000

x24 = -34.5575191895000

x25 = 28.2743338823000

x26 = -31.4159265359000

x27 = -100.530964915000

x28 = -28.2743338823000

x29 = 72.2566310326000

x30 = 56.5486677646000

x31 = -75.3982236862000

x32 = -69.1150383790000

x33 = -6.28318530718000

x34 = -9.42477796077000

x35 = 6.28318530718000

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 = -449.247749463000

x45 = 91.1061869541000

x46 = 59.6902604182000

x47 = -47.1238898038000

x48 = 12.5663706144000

x49 = -62.8318530718000

x50 = 62.8318530718000

x51 = -18.8495559215000

x52 = -12.5663706144000

x53 = -37.6991118431000

x54 = -97.3893722613000

x55 = 94.2477796077000

x56 = 34.5575191895000

x57 = 210.486707791000

x58 = -304.734487398000

x59 = 21.9911485751000

x60 = 37.6991118431000

x61 = 53.4070751110000

x62 = -78.5398163397000

x63 = 0.0

x64 = 43.9822971503000

x65 = -40.8407044967000

x66 = -15.7079632679000

x67 = 47.1238898038000