cos4x=-1/2 (уравнение)

Найду корень уравнения: cos4x=-1/2

Вы ввели

cos(4*x) = -1/2

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

Подробное решение

Дано уравнение
$$\cos{\left(4 x \right)} = - \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
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
$$4 x = \pi n + \frac{2 \pi}{3}$$
$$4 x = \pi n - \frac{\pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{4} + \frac{\pi}{6}$$
$$x_{2} = \frac{\pi n}{4} - \frac{\pi}{12}$$

Быстрый ответ

     pi
x1 = --
     6 

$$x_{1} = \frac{\pi}{6}$$

     pi
x2 = --
     3 

$$x_{2} = \frac{\pi}{3}$$

Сумма и произведение корней

сумма

    pi   pi
0 + -- + --
    6    3 

$$\left(0 + \frac{\pi}{6}\right) + \frac{\pi}{3}$$

=

pi
--
2 

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

произведение

  pi pi
1*--*--
  6  3 

$$\frac{\pi}{3} \cdot 1 \frac{\pi}{6}$$

=

  2
pi 
---
 18

$$\frac{\pi^{2}}{18}$$

Численный ответ

x1 = 8.37758040957278

x2 = 63.8790506229925

x3 = -79.5870138909414

x4 = -83.7758040957278

x5 = 70.162235930172

x6 = 40.317105721069

x7 = -12.0427718387609

x8 = -93.7241808320955

x9 = -75.9218224617533

x10 = 0.523598775598299

x11 = 18.3259571459405

x12 = -79.0634151153431

x13 = -13.6135681655558

x14 = -34.0339204138894

x15 = -31.9395253114962

x16 = -56.025068989018

x17 = -100.007366139275

x18 = 78.0162175641465

x19 = 85.870199198121

x20 = -16.2315620435473

x21 = -82.2050077689329

x22 = 38.2227106186758

x23 = 30.3687289847013

x24 = 46.0766922526503

x25 = 84.2994028713261

x26 = -63.8790506229925

x27 = 74.3510261349584

x28 = -17.8023583703422

x29 = 52.3598775598299

x30 = 9.94837673636768

x31 = -68.0678408277789

x32 = 96.342174710087

x33 = 24.0855436775217

x34 = 41.8879020478639

x35 = -9.94837673636768

x36 = -38.2227106186758

x37 = 68.0678408277789

x38 = -39.7935069454707

x39 = 71.733032256967

x40 = -25.6563400043166

x41 = 12.0427718387609

x42 = -57.5958653158129

x43 = -3.66519142918809

x44 = 49.7418836818384

x45 = 90.0589894029074

x46 = 5.75958653158129

x47 = -53.9306738866248

x48 = 27.7507351067098

x49 = 100.007366139275

x50 = 60.2138591938044

x51 = -47.6474885794452

x52 = 43.4586983746588

x53 = -27.7507351067098

x54 = -61.7846555205993

x55 = -92.1533845053006

x56 = -2.0943951023932

x57 = 75.9218224617533

x58 = 92.1533845053006

x59 = -35.6047167406843

x60 = 88.4881930761125

x61 = -90.0589894029074

x62 = -42.9350995990605

x63 = 48.1710873550435

x64 = 4.18879020478639

x65 = 2.0943951023932

x66 = 31.9395253114962

x67 = 16.2315620435473

x68 = 97.9129710368819

x69 = -5.75958653158129

x70 = -49.7418836818384

x71 = -46.0766922526503

x72 = -24.0855436775217

x73 = 53.9306738866248

x74 = -41.8879020478639

x75 = -60.2138591938044

x76 = -71.733032256967

x77 = 82.2050077689329

x78 = 66.497044500984

x79 = 19.8967534727354

x80 = 34.0339204138894

x81 = 44.5058959258554

x82 = -78.0162175641465

x83 = 93.7241808320955

x84 = -19.8967534727354

x85 = -69.6386371545737

x86 = -97.9129710368819

x87 = 56.025068989018

x88 = 62.3082542961976

x89 = -64.9262481741891

x90 = 26.1799387799149

x91 = -93.2005820564972

x92 = -85.870199198121

x93 = -86.9173967493176

x94 = 22.5147473507269

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