2cos2x-1=0 (уравнение)

Найду корень уравнения: 2cos2x-1=0

Вы ввели

2*cos(2*x) - 1 = 0

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

Подробное решение

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

Перенесём -1 в правую часть ур-ния

с изменением знака при -1

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

Разделим обе части ур-ния на 2

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

Быстрый ответ

     pi
x1 = --
     6 

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

     5*pi
x2 = ----
      6  

$$x_{2} = \frac{5 \pi}{6}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    6     6  

$$\left(0 + \frac{\pi}{6}\right) + \frac{5 \pi}{6}$$

=

pi

$$\pi$$

произведение

  pi 5*pi
1*--*----
  6   6  

$$\frac{5 \pi}{6} \cdot 1 \frac{\pi}{6}$$

=

    2
5*pi 
-----
  36 

$$\frac{5 \pi^{2}}{36}$$

Численный ответ

x1 = -7272.26339428477

x2 = -71.733032256967

x3 = 125.140107367993

x4 = -16.2315620435473

x5 = 0.523598775598299

x6 = 71.733032256967

x7 = -38.2227106186758

x8 = -68.5914396033772

x9 = -31.9395253114962

x10 = -13.0899693899575

x11 = 27.7507351067098

x12 = -18.3259571459405

x13 = 90.5825881785057

x14 = 84.2994028713261

x15 = -2.61799387799149

x16 = -90.5825881785057

x17 = -91.6297857297023

x18 = -25.6563400043166

x19 = 68.5914396033772

x20 = -2685.53812004368

x21 = 74.8746249105567

x22 = -49.7418836818384

x23 = -84.2994028713261

x24 = -34.0339204138894

x25 = -5.75958653158129

x26 = -75.9218224617533

x27 = -47.6474885794452

x28 = -12.0427718387609

x29 = 34.0339204138894

x30 = 85.3466004225227

x31 = 30.8923277602996

x32 = 62.3082542961976

x33 = 31.9395253114962

x34 = -3.66519142918809

x35 = -43.4586983746588

x36 = -87.4409955249159

x37 = -1152.44090509186

x38 = 38.2227106186758

x39 = 96.8657734856853

x40 = 19.3731546971371

x41 = -63.3554518473942

x42 = 12.0427718387609

x43 = 22.5147473507269

x44 = 16.2315620435473

x45 = 82.2050077689329

x46 = -65.4498469497874

x47 = 69.6386371545737

x48 = 60.2138591938044

x49 = 93.7241808320955

x50 = 131.423292675173

x51 = -27.7507351067098

x52 = -97.9129710368819

x53 = 41.3643032722656

x54 = 2.61799387799149

x55 = 53.9306738866248

x56 = 8.90117918517108

x57 = -85.3466004225227

x58 = -40.317105721069

x59 = -60.2138591938044

x60 = 100.007366139275

x61 = 3.66519142918809

x62 = -81.1578102177363

x63 = 66.497044500984

x64 = -217.293491873294

x65 = 24.60914245312

x66 = -62.3082542961976

x67 = -21.4675497995303

x68 = -53.9306738866248

x69 = -78.0162175641465

x70 = -56.025068989018

x71 = 52.8834763354282

x72 = -82.2050077689329

x73 = 56.025068989018

x74 = 5.75958653158129

x75 = 47.6474885794452

x76 = 78.0162175641465

x77 = 75.9218224617533

x78 = -93.7241808320955

x79 = -35.081117965086

x80 = 46.6002910282486

x81 = -41.3643032722656

x82 = 40.317105721069

x83 = 97.9129710368819

x84 = -69.6386371545737

x85 = 91.6297857297023

x86 = 44.5058959258554

x87 = -24.60914245312

x88 = -100.007366139275

x89 = 18.3259571459405

x90 = 88.4881930761125

x91 = 49.7418836818384

x92 = -19.3731546971371

x93 = -9.94837673636768

x94 = 9.94837673636768

x95 = 25.6563400043166

x96 = -46.6002910282486

x97 = 63.3554518473942

График