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Реши любую задачу с помощью нейросети.
$$16 cos^{4}{left (x right )} – 24 cos^{2}{left (x right )} + 9 = 0$$
преобразуем
$$16 cos^{4}{left (x right )} – 24 cos^{2}{left (x right )} + 9 = 0$$
$$16 cos^{4}{left (x right )} – 24 cos^{2}{left (x right )} + 9 = 0$$
Сделаем замену
$$w = cos{left (x right )}$$
Дано уравнение:
$$16 w^{4} – 24 w^{2} + 9 = 0$$
Сделаем замену
$$v = w^{2}$$
тогда ур-ние будет таким:
$$16 v^{2} – 24 v + 9 = 0$$
Это уравнение вида
a*v^2 + b*v + c = 0
Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$v_{1} = frac{sqrt{D} – b}{2 a}$$
$$v_{2} = frac{- sqrt{D} – b}{2 a}$$
где D = b^2 – 4*a*c – это дискриминант.
Т.к.
$$a = 16$$
$$b = -24$$
$$c = 9$$
, то
D = b^2 – 4 * a * c =
(-24)^2 – 4 * (16) * (9) = 0
Т.к. D = 0, то корень всего один.
v = -b/2a = –24/2/(16)
$$v_{1} = frac{3}{4}$$
Получаем окончательный ответ:
Т.к.
$$v = w^{2}$$
то
$$w_{1} = sqrt{v_{1}}$$
$$w_{2} = – sqrt{v_{1}}$$
тогда:
$$w_{1} = $$
2 _____ ___
/ 3/4 / 3
——- = —–
1 2
$$w_{2} = $$
2 _____ ___
-/ 3/4 -/ 3
——— = ——-
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:
$$x_{1} = pi n + {acos}{left (w_{1} right )}$$
$$x_{1} = pi n + {acos}{left (frac{3}{4} right )}$$
$$x_{1} = pi n + {acos}{left (frac{3}{4} right )}$$
$$x_{2} = pi n + {acos}{left (w_{1} right )} – pi$$
$$x_{2} = pi n – pi + {acos}{left (frac{3}{4} right )}$$
$$x_{2} = pi n – pi + {acos}{left (frac{3}{4} right )}$$
pi
x1 = —
6
5*pi
x2 = —-
6
7*pi
x3 = —-
6
11*pi
x4 = —–
6
x1 = -63.3554520784000
x2 = 31.9395253720000
x3 = 82.2050078121000
x4 = -12.0427717941000
x5 = 25.6563402381000
x6 = -43.4586981070000
x7 = -35.0811180699000
x8 = 46.6002908137000
x9 = 78.0162175404000
x10 = 75.9218225371000
x11 = 38.2227106421000
x12 = -13.0899695144000
x13 = 3.66519165966000
x14 = -12.0427716933000
x15 = 40.3171056733000
x16 = -5.75958652411000
x17 = -57.0722666229000
x18 = 96.8657732925000
x19 = -78.0162177480000
x20 = 71.7330321984000
x21 = 49.7418836562000
x22 = 47.6474888164000
x23 = -25.6563400702000
x24 = -56.0250689615000
x25 = -38.2227105836000
x26 = -90.5825879539000
x27 = 30.8923276031000
x28 = -69.6386372324000
x29 = 63.3554520012000
x30 = -79.0634151733000
x31 = 0.523598869090000
x32 = 74.8746247279000
x33 = -100.007366131000
x34 = -9.94837688854000
x35 = 93.7241807252000
x36 = 69.6386373945000
x37 = 18.3259570921000
x38 = 53.9306739547000
x39 = 12.0427717389000
x40 = -40.3171054977000
x41 = 85.3466005604000
x42 = -62.3082540767000
x43 = 16.2315620586000
x44 = 22.5147474000000
x45 = 5.75958654549000
x46 = 90.5825879717000
x47 = 91.6297857458000
x48 = -93.7241808491000
x49 = -60.2138591519000
x50 = -24.6091422517000
x51 = -46.6002908181000
x52 = -82.2050077217000
x53 = -2.61799368636000
x54 = -68.5914393855000
x55 = -79.0634152777000
x56 = 66.4970444990000
x57 = 34.0339203481000
x58 = -18.3259569189000
x59 = -34.0339203776000
x60 = -41.3643035003000
x61 = 52.8834761647000
x62 = -53.9306740025000
x63 = 97.9129711194000
x64 = 100.007366130000
x65 = 88.4881930588000
x66 = 9.94837678906000
x67 = -91.6297858134000
x68 = -22.5147476522000
x69 = 91.6297859724000
x70 = -78.0162175458000
x71 = -16.2315620175000
x72 = 68.5914393927000
x73 = 8.90117904331000
x74 = 60.2138592265000
x75 = -31.9395255525000
x76 = 44.5058959450000
x77 = -27.7507351062000
x78 = -3.66519148908000
x79 = -85.3466006562000
x80 = 56.0250689473000
x81 = -71.7330322686000
x82 = -47.6474886513000
x83 = 27.7507351041000
x84 = -84.2994026557000
x85 = -49.7418836876000
x86 = -21.4675496384000
x87 = 19.3731548777000
x88 = 62.3082542546000
x89 = 24.6091422350000
x90 = 41.3643034403000
x91 = -19.3731549220000
x92 = 84.2994028359000
x93 = 2.61799365640000
