На странице представлен фрагмент
Реши любую задачу с помощью нейросети.
$$4 cos^{2}{left (x right )} = 3$$
преобразуем
$$4 cos^{2}{left (x right )} – 3 = 0$$
$$4 cos^{2}{left (x right )} – 3 = 0$$
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.
$$a = 4$$
$$b = 0$$
$$c = -3$$
, то
D = b^2 – 4 * a * c =
(0)^2 – 4 * (4) * (-3) = 48
Т.к. D > 0, то уравнение имеет два корня.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b – sqrt(D)) / (2*a)
или
$$w_{1} = frac{sqrt{3}}{2}$$
$$w_{2} = – frac{sqrt{3}}{2}$$
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:
$$x_{1} = pi n + {acos}{left (w_{1} right )}$$
$$x_{1} = pi n + {acos}{left (frac{sqrt{3}}{2} right )}$$
$$x_{1} = pi n + frac{pi}{6}$$
$$x_{2} = pi n + {acos}{left (w_{2} right )}$$
$$x_{2} = pi n + {acos}{left (- frac{sqrt{3}}{2} right )}$$
$$x_{2} = pi n + frac{5 pi}{6}$$
$$x_{3} = pi n + {acos}{left (w_{1} right )} – pi$$
$$x_{3} = pi n – pi + {acos}{left (frac{sqrt{3}}{2} right )}$$
$$x_{3} = pi n – frac{5 pi}{6}$$
$$x_{4} = pi n + {acos}{left (w_{2} right )} – pi$$
$$x_{4} = pi n – pi + {acos}{left (- frac{sqrt{3}}{2} right )}$$
$$x_{4} = pi n – frac{pi}{6}$$
pi
x1 = —
6
5*pi
x2 = —-
6
7*pi
x3 = —-
6
11*pi
x4 = —–
6
x1 = -3.66519142919000
x2 = -100.007366139000
x3 = 71.7330322570000
x4 = 41.3643032723000
x5 = 88.4881930761000
x6 = -69.6386371546000
x7 = -5.75958653158000
x8 = -21.4675497995000
x9 = -60.2138591938000
x10 = -2.61799387799000
x11 = -43.4586983747000
x12 = -71.7330322570000
x13 = -85.3466004225000
x14 = -24.6091424531000
x15 = -12.0427718388000
x16 = 52.8834763354000
x17 = -41.3643032723000
x18 = 31.9395253115000
x19 = 25.6563400043000
x20 = 18.3259571459000
x21 = -13.0899693900000
x22 = 63.3554518474000
x23 = -47.6474885794000
x24 = 0.523598775598000
x25 = 19.3731546971000
x26 = -31.9395253115000
x27 = -25.6563400043000
x28 = -75.9218224618000
x29 = -62.3082542962000
x30 = -27.7507351067000
x31 = -40.3171057211000
x32 = 62.3082542962000
x33 = 22.5147473507000
x34 = 49.7418836818000
x35 = 125.140107368000
x36 = 46.6002910282000
x37 = 5.75958653158000
x38 = -68.5914396034000
x39 = 66.4970445010000
x40 = 97.9129710369000
x41 = 75.9218224618000
x42 = -16.2315620435000
x43 = 12.0427718388000
x44 = -313.635666583000
x45 = -34.0339204139000
x46 = -97.9129710369000
x47 = 78.0162175641000
x48 = -2686.58531759000
x49 = -91.6297857297000
x50 = 16.2315620435000
x51 = 100.007366139000
x52 = 2.61799387799000
x53 = 84.2994028713000
x54 = 69.6386371546000
x55 = -49.7418836818000
x56 = -93.7241808321000
x57 = 9.94837673637000
x58 = -90.5825881785000
x59 = -46.6002910282000
x60 = 47.6474885794000
x61 = -63.3554518474000
x62 = -87.4409955249000
x63 = 68.5914396034000
x64 = -65.4498469498000
x65 = -56.0250689890000
x66 = -18.3259571459000
x67 = -82.2050077689000
x68 = 34.0339204139000
x69 = 85.3466004225000
x70 = 30.8923277603000
x71 = -217.293491873000
x72 = 24.6091424531000
x73 = 8.90117918517000
x74 = 91.6297857297000
x75 = 82.2050077689000
x76 = 74.8746249106000
x77 = -35.0811179651000
x78 = 44.5058959259000
x79 = -78.0162175641000
x80 = -9.94837673637000
x81 = -81.1578102177000
x82 = 38.2227106187000
x83 = -38.2227106187000
x84 = 40.3171057211000
x85 = -84.2994028713000
x86 = -53.9306738866000
x87 = 93.7241808321000
x88 = 90.5825881785000
x89 = 53.9306738866000
x90 = 56.0250689890000
x91 = 3.66519142919000
x92 = 131.423292675000
x93 = 27.7507351067000
x94 = 60.2138591938000
x95 = -19.3731546971000
x96 = 96.8657734857000
