2cos*6x=1 (уравнение)

Найду корень уравнения: 2cos*6x=1

Вы ввели

2*cos(6*x) = 1

$$2 \cos{\left(6 x \right)} = 1$$

Подробное решение

Дано уравнение
$$2 \cos{\left(6 x \right)} = 1$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на 2

Ур-ние превратится в
$$\cos{\left(6 x \right)} = \frac{1}{2}$$
Это ур-ние преобразуется в
$$6 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$6 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Или
$$6 x = \pi n + \frac{\pi}{3}$$
$$6 x = \pi n - \frac{2 \pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$6$$
получим ответ:
$$x_{1} = \frac{\pi n}{6} + \frac{\pi}{18}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{9}$$

Быстрый ответ

     pi
x1 = --
     18

$$x_{1} = \frac{\pi}{18}$$

     5*pi
x2 = ----
      18 

$$x_{2} = \frac{5 \pi}{18}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    18    18 

$$\left(0 + \frac{\pi}{18}\right) + \frac{5 \pi}{18}$$

=

pi
--
3 

$$\frac{\pi}{3}$$

произведение

  pi 5*pi
1*--*----
  18  18 

$$\frac{5 \pi}{18} \cdot 1 \frac{\pi}{18}$$

=

    2
5*pi 
-----
 324 

$$\frac{5 \pi^{2}}{324}$$

Численный ответ

x1 = -32.2885911618951

x2 = -47.9965544298441

x3 = -6.10865238198015

x4 = -99.6583002888762

x5 = -72.4311639577647

x6 = 30.1941960595019

x7 = -83.9503370209273

x8 = -59.8647933434055

x9 = 20.0712863979348

x10 = -57.7703982410123

x11 = 96.1676417848876

x12 = -65.7989128001862

x13 = -23.9110107523223

x14 = 44.1568300754565

x15 = -4.01425727958696

x16 = -21.8166156499291

x17 = 15.8824961931484

x18 = -39.9680398706701

x19 = 6.10865238198015

x20 = -87.7900613753148

x21 = -10.2974425867665

x22 = -36.4773813666815

x23 = 83.9503370209273

x24 = -17.9768912955416

x25 = -67.8933079025794

x26 = 8.20304748437335

x27 = 76.2708883121522

x28 = -126.885436619988

x29 = 4.01425727958696

x30 = -26.0054058547155

x31 = 86.0447321233204

x32 = 28.0998009571087

x33 = 37.873644768277

x34 = 66.1479786505851

x35 = 22.165681500328

x36 = -13.7881010907552

x37 = -43.8077642250577

x38 = 56.3741348394168

x39 = -45.9021593274509

x40 = 12.3918376891597

x41 = 69.9877030049726

x42 = 81.8559419185341

x43 = -35.7792496658838

x44 = -41.7133691226645

x45 = -11.693705988362

x46 = -69.9877030049726

x47 = -86.0447321233204

x48 = 24.2600766027212

x49 = -77.6671517137477

x50 = -94.0732466824944

x51 = -28.0998009571087

x52 = -81.8559419185341

x53 = 32.2885911618951

x54 = 94.0732466824944

x55 = 34.3829862642883

x56 = -37.873644768277

x57 = -61.9591884457987

x58 = 42.0624349730633

x59 = 74.176493209759

x60 = -15.8824961931484

x61 = 64.0535835481919

x62 = -89.884456477708

x63 = 90.2335223281068

x64 = 47.9965544298441

x65 = -1.91986217719376

x66 = 54.2797397370236

x67 = -50.0909495322373

x68 = -53.2325421858271

x69 = -19.7222205475359

x70 = -72.0820981073658

x71 = 39.9680398706701

x72 = -96.5167076352864

x73 = 98.2620368872808

x74 = 61.9591884457987

x75 = 72.0820981073658

x76 = 88.1391272257137

x77 = 10.2974425867665

x78 = -55.6760031386191

x79 = 68.2423737529783

x80 = 2.26892802759263

x81 = -79.7615468161409

x82 = -33.6848545634906

x83 = 50.0909495322373

x84 = 91.9788515801012

x85 = 26.0054058547155

x86 = 0.174532925199433

x87 = 59.8647933434055

x88 = -91.9788515801012

x89 = 26.3544717051144

x90 = -31.2413936106985

x91 = -58.46852994181

x92 = 46.2512251778497

x93 = 100.356431989674

x94 = 52.1853446346305

x95 = 17.9768912955416

x96 = 78.3652834145454

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