cos(5x)=-1/2 (уравнение)

Найду корень уравнения: cos(5x)=-1/2

Вы ввели

cos(5*x) = -1/2

$$\cos{\left(5 x \right)} = - \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(5 x \right)} = - \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$5 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$5 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Или
$$5 x = \pi n + \frac{2 \pi}{3}$$
$$5 x = \pi n - \frac{\pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$5$$
получим ответ:
$$x_{1} = \frac{\pi n}{5} + \frac{2 \pi}{15}$$
$$x_{2} = \frac{\pi n}{5} - \frac{\pi}{15}$$

Быстрый ответ

     2*pi
x1 = ----
      15 

$$x_{1} = \frac{2 \pi}{15}$$

     4*pi
x2 = ----
      15 

$$x_{2} = \frac{4 \pi}{15}$$

Сумма и произведение корней

сумма

    2*pi   4*pi
0 + ---- + ----
     15     15 

$$\left(0 + \frac{2 \pi}{15}\right) + \frac{4 \pi}{15}$$

=

2*pi
----
 5  

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

произведение

  2*pi 4*pi
1*----*----
   15   15 

$$\frac{4 \pi}{15} \cdot 1 \cdot \frac{2 \pi}{15}$$

=

    2
8*pi 
-----
 225 

$$\frac{8 \pi^{2}}{225}$$

Численный ответ

x1 = -99.6932068739161

x2 = 58.2241838465308

x3 = -29.7404104539834

x4 = 15.9174027781883

x5 = 24.2949831877611

x6 = 76.2359817271123

x7 = -44.8200551912144

x8 = -83.7758040957278

x9 = -24.2949831877611

x10 = 70.79055446089

x11 = 51.9409985393512

x12 = 98.4365698124802

x13 = 78.3303768295055

x14 = -93.8289005872152

x15 = -97.5988117715229

x16 = -55.710909723659

x17 = 46.0766922526503

x18 = -63.6696111127531

x19 = 28.0648943720688

x20 = 95.0855376486511

x21 = -73.7227076042405

x22 = 44.4011761707357

x23 = 68.2772803380182

x24 = -38.1179908635562

x25 = -49.8466034369581

x26 = 61.9940950308386

x27 = 66.182885235625

x28 = 72.0471915223259

x29 = 59.8996999284454

x30 = 36.023595761163

x31 = -61.9940950308386

x32 = -9.6342174710087

x33 = -80.0058929114201

x34 = 14.2418866962737

x35 = -15.9174027781883

x36 = -39.7935069454707

x37 = 56.1297887441376

x38 = 22.2005880853679

x39 = 38.1179908635562

x40 = 90.0589894029074

x41 = -18.0117978805815

x42 = 32.2536845768552

x43 = -48.1710873550435

x44 = -82.1002880138133

x45 = -95.9232956896084

x46 = 7.95870138909414

x47 = 83.3569250752492

x48 = -28.0648943720688

x49 = 23.0383461263252

x50 = 433.95866521587

x51 = -51.9409985393512

x52 = 88.3834733209928

x53 = 28.4837733925475

x54 = 0.418879020478639

x55 = -7.95870138909414

x56 = 93.8289005872152

x57 = -92.1533845053006

x58 = -21.7817090648892

x59 = 10.8908545324446

x60 = 92.1533845053006

x61 = -4.18879020478639

x62 = 18.0117978805815

x63 = -25.9704992696756

x64 = 42.3067810683426

x65 = 100.112085894395

x66 = -71.2094334813686

x67 = -90.0589894029074

x68 = -43.5634181297785

x69 = -58.6430628670095

x70 = 48.1710873550435

x71 = 4.18879020478639

x72 = 5.86430628670095

x73 = 2.0943951023932

x74 = 34.3480796792484

x75 = 48.5899663755221

x76 = -36.023595761163

x77 = -53.6165146212658

x78 = 69.9527964199327

x79 = -41.8879020478639

x80 = 82.1002880138133

x81 = 49.8466034369581

x82 = -5.86430628670095

x83 = -59.8996999284454

x84 = -31.8348055563766

x85 = -19.687313962496

x86 = -69.9527964199327

x87 = -72.0471915223259

x88 = -77.4926187885482

x89 = 86.2890782185997

x90 = -11.7286125734019

x91 = 25.9704992696756

x92 = -27302.1156757772

x93 = -75.8171027066337

x94 = -85.870199198121

x95 = -65.7640062151463

x96 = 80.0058929114201

x97 = 12.1474915938805

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