2cos8x+1=0 (уравнение)

Найду корень уравнения: 2cos8x+1=0

Вы ввели

2*cos(8*x) + 1 = 0

$$2 \cos{\left(8 x \right)} + 1 = 0$$

Подробное решение

Дано уравнение
$$2 \cos{\left(8 x \right)} + 1 = 0$$
- это простейшее тригонометрическое ур-ние

Перенесём 1 в правую часть ур-ния

с изменением знака при 1

Получим:
$$2 \cos{\left(8 x \right)} = -1$$

Разделим обе части ур-ния на 2

Ур-ние превратится в
$$\cos{\left(8 x \right)} = - \frac{1}{2}$$
Это ур-ние преобразуется в
$$8 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$8 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Или
$$8 x = \pi n + \frac{2 \pi}{3}$$
$$8 x = \pi n - \frac{\pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$8$$
получим ответ:
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{24}$$

Быстрый ответ

     pi
x1 = --
     12

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

     pi
x2 = --
     6 

$$x_{2} = \frac{\pi}{6}$$

Сумма и произведение корней

сумма

    pi   pi
0 + -- + --
    12   6 

$$\left(0 + \frac{\pi}{12}\right) + \frac{\pi}{6}$$

=

pi
--
4 

$$\frac{\pi}{4}$$

произведение

  pi pi
1*--*--
  12 6 

$$\frac{\pi}{6} \cdot 1 \frac{\pi}{12}$$

=

  2
pi 
---
 72

$$\frac{\pi^{2}}{72}$$

Численный ответ

x1 = 6.02138591938044

x2 = -87.7027949127151

x3 = -43.720497762458

x4 = -83.7758040957278

x5 = 70.162235930172

x6 = 100.269165527074

x7 = -52.3598775598299

x8 = 40.317105721069

x9 = -12.0427718387609

x10 = -93.7241808320955

x11 = 18.3259571459405

x12 = -96.0803753222878

x13 = 96.0803753222878

x14 = 66.2352451131848

x15 = -34.0339204138894

x16 = 22.2529479629277

x17 = -35.8665161284835

x18 = -56.025068989018

x19 = -100.007366139275

x20 = -37.9609112308767

x21 = 38.2227106186758

x22 = 46.0766922526503

x23 = 84.2994028713261

x24 = -89.7971900151083

x25 = 0.261799387799149

x26 = 74.0892267471593

x27 = -63.8790506229925

x28 = -17.8023583703422

x29 = 44.2440965380563

x30 = -45.8148928648512

x31 = 9.94837673636768

x32 = 24.0855436775217

x33 = 78.2780169519457

x34 = 28.012534494509

x35 = -1.83259571459405

x36 = 68.0678408277789

x37 = -39.7935069454707

x38 = -67.8060414399797

x39 = -13.8753675533549

x40 = 64.1408500107916

x41 = 56.2868683768171

x42 = -3.66519142918809

x43 = 90.0589894029074

x44 = 52.0980781720307

x45 = -11.2573736753634

x46 = -57.857664703612

x47 = 12.30457122656

x48 = 88.2263936883134

x49 = 139.539073696947

x50 = 81.9432083811338

x51 = 86.1319985859202

x52 = -90.8443875663049

x53 = 60.2138591938044

x54 = 8.11578102177363

x55 = -78.801615727544

x56 = -27.7507351067098

x57 = -61.7846555205993

x58 = 93.9859802198946

x59 = 75.9218224617533

x60 = 92.1533845053006

x61 = -15.9697626557481

x62 = 42.1497014356631

x63 = 30.1069295969022

x64 = -34.2957198016886

x65 = -25.3945406165175

x66 = 48.1710873550435

x67 = 4.18879020478639

x68 = -79.8488132787406

x69 = 2.0943951023932

x70 = 71.9948316447661

x71 = 31.9395253114962

x72 = 16.2315620435473

x73 = 97.9129710368819

x74 = -5.75958653158129

x75 = -49.7418836818384

x76 = 53.9306738866248

x77 = -41.8879020478639

x78 = -71.733032256967

x79 = 82.2050077689329

x80 = -59.9520598060052

x81 = 50.0036830696375

x82 = -23.8237442897226

x83 = -65.7116463375865

x84 = 34.2957198016886

x85 = -78.0162175641465

x86 = -19.8967534727354

x87 = -97.9129710368819

x88 = 62.3082542961976

x89 = -21.7293491873294

x90 = 26.1799387799149

x91 = -81.9432083811338

x92 = 20.1585528605345

x93 = -85.870199198121

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