2cos4x=1 (уравнение)

Найду корень уравнения: 2cos4x=1

Вы ввели

2*cos(4*x) = 1

$$2 \cos{\left(4 x \right)} = 1$$

Подробное решение

Дано уравнение
$$2 \cos{\left(4 x \right)} = 1$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на 2

Ур-ние превратится в
$$\cos{\left(4 x \right)} = \frac{1}{2}$$
Это ур-ние преобразуется в
$$4 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$4 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Или
$$4 x = \pi n + \frac{\pi}{3}$$
$$4 x = \pi n - \frac{2 \pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{4} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{4} - \frac{\pi}{6}$$

Быстрый ответ

     pi
x1 = --
     12

$$x_{1} = \frac{\pi}{12}$$

     5*pi
x2 = ----
      12 

$$x_{2} = \frac{5 \pi}{12}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    12    12 

$$\left(0 + \frac{\pi}{12}\right) + \frac{5 \pi}{12}$$

=

pi
--
2 

$$\frac{\pi}{2}$$

произведение

  pi 5*pi
1*--*----
  12  12 

$$\frac{5 \pi}{12} \cdot 1 \frac{\pi}{12}$$

=

    2
5*pi 
-----
 144 

$$\frac{5 \pi^{2}}{144}$$

Численный ответ

x1 = 6.02138591938044

x2 = -87.7027949127151

x3 = -42.1497014356631

x4 = -43.720497762458

x5 = 100.269165527074

x6 = -100.269165527074

x7 = 45.8148928648512

x8 = -96.0803753222878

x9 = 96.0803753222878

x10 = 66.2352451131848

x11 = 22.2529479629277

x12 = -35.8665161284835

x13 = -37.9609112308767

x14 = -39.5317075576716

x15 = -89.7971900151083

x16 = 0.261799387799149

x17 = 74.0892267471593

x18 = -28.012534494509

x19 = 4.45058959258554

x20 = 92.9387826686981

x21 = 44.2440965380563

x22 = -45.8148928648512

x23 = -45.2912940892529

x24 = 78.2780169519457

x25 = 26.4417381677141

x26 = 126.972703082587

x27 = 28.012534494509

x28 = -1.83259571459405

x29 = 23.8237442897226

x30 = 70.4240353179712

x31 = -67.8060414399797

x32 = -13.8753675533549

x33 = -31.6777259236971

x34 = -34.8193185772869

x35 = 64.1408500107916

x36 = -97.6511716490827

x37 = 56.2868683768171

x38 = 52.0980781720307

x39 = 1448.53601269269

x40 = -57.857664703612

x41 = 48.4328867428426

x42 = 12.30457122656

x43 = 88.2263936883134

x44 = -64.6644487863899

x45 = 62.5700536839967

x46 = 468.359104772678

x47 = 81.9432083811338

x48 = 86.1319985859202

x49 = 89.7971900151083

x50 = -86.1319985859202

x51 = 92.4151838930998

x52 = -71.9948316447661

x53 = 8.11578102177363

x54 = 93.9859802198946

x55 = -83.5140047079287

x56 = 87.7027949127151

x57 = -9.68657734856853

x58 = 42.1497014356631

x59 = -15.9697626557481

x60 = -17.540558982543

x61 = 59.9520598060052

x62 = 30.1069295969022

x63 = -93.9859802198946

x64 = -78.2780169519457

x65 = -61.5228561328001

x66 = 9.16297857297023

x67 = -50.0036830696375

x68 = -79.8488132787406

x69 = 71.9948316447661

x70 = -53.6688744988256

x71 = -52.0980781720307

x72 = -12.8281700021583

x73 = -64.1408500107916

x74 = -75.6600230739542

x75 = -20.1585528605345

x76 = 15.9697626557481

x77 = -59.9520598060052

x78 = 50.0036830696375

x79 = 37.9609112308767

x80 = -23.8237442897226

x81 = -65.7116463375865

x82 = 34.2957198016886

x83 = -74.0892267471593

x84 = 67.8060414399797

x85 = 1.83259571459405

x86 = 85.0848010347236

x87 = -21.7293491873294

x88 = -81.9432083811338

x89 = 20.1585528605345

x90 = -30.1069295969022

x91 = -6.02138591938044

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