2*cos(2*x)+4*sqrt(3)*cos(x)-7=0

$$4 sqrt{3} cos{left (x right )} + 2 cos{left (2 x right )} – 7 = 0$$

Дано уравнение
$$4 sqrt{3} cos{left (x right )} + 2 cos{left (2 x right )} – 7 = 0$$
преобразуем
$$4 cos^{2}{left (x right )} + 4 sqrt{3} cos{left (x right )} – 9 = 0$$
$$4 cos^{2}{left (x right )} + 4 sqrt{3} cos{left (x right )} – 9 = 0$$
Сделаем замену
$$w = cos{left (x right )}$$
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.
$$a = 4$$
$$b = 4 sqrt{3}$$
$$c = -9$$
, то

D = b^2 – 4 * a * c =

(4*sqrt(3))^2 – 4 * (4) * (-9) = 192

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

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

w2 = (-b – sqrt(D)) / (2*a)

или
$$w_{1} = frac{sqrt{3}}{2}$$
$$w_{2} = – frac{3 sqrt{3}}{2}$$
делаем обратную замену
$$cos{left (x right )} = w$$
Дано уравнение
$$cos{left (x right )} = w$$
– это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
Или
$$x = pi n + {acos}{left (w right )}$$
$$x = pi n + {acos}{left (w right )} – pi$$
, где n – любое целое число
подставляем w:
$$x_{1} = pi n + {acos}{left (w_{1} right )}$$
$$x_{1} = pi n + {acos}{left (frac{sqrt{3}}{2} right )}$$
$$x_{1} = pi n + frac{pi}{6}$$
$$x_{2} = pi n + {acos}{left (w_{2} right )}$$
$$x_{2} = pi n + {acos}{left (- frac{3 sqrt{3}}{2} right )}$$
$$x_{2} = pi n + {acos}{left (- frac{3 sqrt{3}}{2} right )}$$
$$x_{3} = pi n + {acos}{left (w_{1} right )} – pi$$
$$x_{3} = pi n – pi + {acos}{left (frac{sqrt{3}}{2} right )}$$
$$x_{3} = pi n – frac{5 pi}{6}$$
$$x_{4} = pi n + {acos}{left (w_{2} right )} – pi$$
$$x_{4} = pi n – pi + {acos}{left (- frac{3 sqrt{3}}{2} right )}$$
$$x_{4} = pi n – pi + {acos}{left (- frac{3 sqrt{3}}{2} right )}$$

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

/ ___
|I / 3 |
x2 = -I*log|- + —–|
2 2 /

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

/ 2
x3 = pi + I*log|——————|
| ____ ___|
– / 23 + 3*/ 3 /

$$x_{3} = pi + i log{left (frac{2}{- sqrt{23} + 3 sqrt{3}} right )}$$

/ 2
x4 = pi + I*log|—————-|
| ____ ___|
/ 23 + 3*/ 3 /

$$x_{4} = pi + i log{left (frac{2}{sqrt{23} + 3 sqrt{3}} right )}$$

x1 = -88.4881930761000

x2 = -30.8923277603000

x3 = 44.5058959259000

x4 = 100.007366139000

x5 = -49.7418836818000

x6 = 37.1755130675000

x7 = 69.6386371546000

x8 = 88.4881930761000

x9 = -93.7241808321000

x10 = -69.6386371546000

x11 = 62.3082542962000

x12 = 38.2227106187000

x13 = -100.007366139000

x14 = 57.0722665402000

x15 = 49.7418836818000

x16 = -38.2227106187000

x17 = -62.3082542962000

x18 = -43.4586983747000

x19 = 696.909970321000

x20 = -57.0722665402000

x21 = -101.054563690000

x22 = -24.6091424531000

x23 = -87.4409955249000

x24 = -74.8746249106000

x25 = 68.5914396034000

x26 = 93.7241808321000

x27 = 5.75958653158000

x28 = -68.5914396034000

x29 = -50.7890812330000

x30 = -12.0427718388000

x31 = -94.7713783833000

x32 = -63.3554518474000

x33 = 13.0899693900000

x34 = 31.9395253115000

x35 = -56.0250689890000

x36 = -760.789020944000

x37 = -18.3259571459000

x38 = -5.75958653158000

x39 = 75.9218224618000

x40 = -82.2050077689000

x41 = 30.8923277603000

x42 = 25.6563400043000

x43 = 18.3259571459000

x44 = -176.452787377000

x45 = -13.0899693900000

x46 = 213.104701669000

x47 = 81.1578102177000

x48 = 63.3554518474000

x49 = -5159.01873597000

x50 = 56.0250689890000

x51 = -25.6563400043000

x52 = 0.523598775598000

x53 = 19.3731546971000

x54 = 24.6091424531000

x55 = -6.80678408278000

x56 = 169.122404518000

x57 = 12.0427718388000

x58 = -19.3731546971000

x59 = -31.9395253115000

x60 = 82.2050077689000

x61 = 74.8746249106000

x62 = -75.9218224618000