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Реши любую задачу с помощью нейросети.
$$6 cos^{2}{left (x right )} + cos{left (x right )} – 1 = 0$$
преобразуем
$$6 cos^{2}{left (x right )} + cos{left (x right )} – 1 = 0$$
$$6 cos^{2}{left (x right )} + cos{left (x right )} – 1 = 0$$
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.
$$a = 6$$
$$b = 1$$
$$c = -1$$
, то
D = b^2 – 4 * a * c =
(1)^2 – 4 * (6) * (-1) = 25
Т.к. D > 0, то уравнение имеет два корня.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b – sqrt(D)) / (2*a)
или
$$w_{1} = frac{1}{3}$$
$$w_{2} = – frac{1}{2}$$
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:
$$x_{1} = pi n + {acos}{left (w_{1} right )}$$
$$x_{1} = pi n + {acos}{left (frac{1}{3} right )}$$
$$x_{1} = pi n + {acos}{left (frac{1}{3} right )}$$
$$x_{2} = pi n + {acos}{left (w_{2} right )}$$
$$x_{2} = pi n + {acos}{left (- frac{1}{2} right )}$$
$$x_{2} = pi n + frac{2 pi}{3}$$
$$x_{3} = pi n + {acos}{left (w_{1} right )} – pi$$
$$x_{3} = pi n – pi + {acos}{left (frac{1}{3} right )}$$
$$x_{3} = pi n – pi + {acos}{left (frac{1}{3} right )}$$
$$x_{4} = pi n + {acos}{left (w_{2} right )} – pi$$
$$x_{4} = pi n – pi + {acos}{left (- frac{1}{2} right )}$$
$$x_{4} = pi n – frac{pi}{3}$$
2*pi
x1 = —-
3
4*pi
x2 = —-
3
x3 = -acos(1/3) + 2*pi
x4 = acos(1/3)
x1 = -93.0168201904000
x2 = 57.7796271820000
x3 = 79.5870138909000
x4 = 48.1710873550000
x5 = 52.3598775598000
x6 = -30.1849671186000
x7 = 26.3637006461000
x8 = -71.2094334814000
x9 = -64.0628124891000
x10 = 92.1533845053000
x11 = 67.8840789616000
x12 = -26.3637006461000
x13 = 80.4504495760000
x14 = -17.6185965042000
x15 = -2.09439510239000
x16 = -33.5103216383000
x17 = 39.7935069455000
x18 = -99.3000054975000
x19 = 77.4926187885000
x20 = -92.1533845053000
x21 = 4.18879020479000
x22 = -23.9017818114000
x23 = -90.0589894029000
x24 = -70.3459977963000
x25 = -4.18879020479000
x26 = 41.8879020479000
x27 = -5.05222588984000
x28 = 32.6468859532000
x29 = 70.3459977963000
x30 = -77.4926187885000
x31 = 8.37758040957000
x32 = 74.1672642688000
x33 = 54.4542726622000
x34 = -51.4964418748000
x35 = -27.2271363311000
x36 = -340.522966005000
x37 = -23.0383461263000
x38 = 17.6185965042000
x39 = -46.0766922527000
x40 = 30.1849671186000
x41 = 83.7758040957000
x42 = 76.6291831035000
x43 = 55.3177083473000
x44 = -82.9123684107000
x45 = 20.0805153389000
x46 = -13.7973300317000
x47 = -20.0805153389000
x48 = 13.7973300317000
x49 = -10.4719755120000
x50 = 99.3000054975000
x51 = 237.530082255000
x52 = -41.8879020479000
x53 = -57.7796271820000
x54 = 60.7374579694000
x55 = 46.0766922527000
x56 = 90.0589894029000
x57 = 85.8701991981000
x58 = 96.3421747101000
x59 = -98.4365698125000
x60 = 36.4681524257000
x61 = 33.5103216383000
x62 = -49.0345230401000
x63 = -1.23095941734000
x64 = 98.4365698125000
x65 = -55.3177083473000
x66 = 10.4719755120000
x67 = 38.9300712604000
x68 = -79.5870138909000
x69 = -11.3354111970000
x70 = -83.7758040957000
x71 = -7.51414472452000
x72 = 61.6008936545000
x73 = -74.1672642688000
x74 = -39.7935069455000
x75 = -32.6468859532000
x76 = 2.09439510239000
x77 = -85.8701991981000
x78 = -61.6008936545000
x79 = 64.0628124891000
x80 = -67.8840789616000
x81 = 82.9123684107000
x82 = 11.3354111970000
x83 = -48.1710873550000
x84 = 23.9017818114000
x85 = -35.6047167407000
x86 = -54.4542726622000
x87 = 16.7551608191000
x88 = -76.6291831035000
x89 = -506.843614779000
