cos(9x)=-1/5 (уравнение)

Найду корень уравнения: cos(9x)=-1/5

Вы ввели

cos(9*x) = -1/5

$$\cos{\left(9 x \right)} = - \frac{1}{5}$$

Подробное решение

Дано уравнение
$$\cos{\left(9 x \right)} = - \frac{1}{5}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$9 x = \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$9 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
Или
$$9 x = \pi n + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
$$9 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{5} \right)}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$9$$
получим ответ:
$$x_{1} = \frac{\pi n}{9} + \frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9}$$
$$x_{2} = \frac{\pi n}{9} - \frac{\pi}{9} + \frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9}$$

Быстрый ответ

       acos(-1/5)   2*pi
x1 = - ---------- + ----
           9         9  

$$x_{1} = - \frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9} + \frac{2 \pi}{9}$$

     acos(-1/5)
x2 = ----------
         9     

$$x_{2} = \frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9}$$

Сумма и произведение корней

сумма

      acos(-1/5)   2*pi   acos(-1/5)
0 + - ---------- + ---- + ----------
          9         9         9     

$$\frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9} + \left(0 + \left(- \frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9} + \frac{2 \pi}{9}\right)\right)$$

=

2*pi
----
 9  

$$\frac{2 \pi}{9}$$

произведение

  /  acos(-1/5)   2*pi\ acos(-1/5)
1*|- ---------- + ----|*----------
  \      9         9  /     9     

$$1 \left(- \frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9} + \frac{2 \pi}{9}\right) \frac{\operatorname{acos}{\left(- \frac{1}{5} \right)}}{9}$$

=

(-acos(-1/5) + 2*pi)*acos(-1/5)
-------------------------------
               81              

$$\frac{\left(- \operatorname{acos}{\left(- \frac{1}{5} \right)} + 2 \pi\right) \operatorname{acos}{\left(- \frac{1}{5} \right)}}{81}$$

Численный ответ

x1 = -29.822757106793

x2 = 52.1629715323204

x3 = -45.8797862251408

x4 = -17.9545181932316

x5 = -77.9938444618365

x6 = -13.7657279884452

x7 = 12.0651449410709

x8 = -53.9530469889348

x9 = 2.29130112990267

x10 = -3.99188417727692

x11 = 17.9545181932316

x12 = 58.1418371937212

x13 = 77.9938444618365

x14 = -65.8212859024963

x15 = 88.1615003280237

x16 = 22.143308398018

x17 = 76.2932614144622

x18 = 24.2377035004111

x19 = -56.047442091328

x20 = 33.0090959650029

x21 = -31.9171522091862

x22 = -71.7106591546569

x23 = -9.97074983867772

x24 = 94.0508735801843

x25 = -67.9156810048895

x26 = -51.8586518865416

x27 = -91.9564784777911

x28 = -89.8620833753979

x29 = 44.1792031777666

x30 = -97.9353441391919

x31 = 30.2165691618119

x32 = 6.08627927967012

x33 = -57.7480251387023

x34 = 61.9368153434887

x35 = 56.047442091328

x36 = -5.78195963389132

x37 = -87.7676882730047

x38 = -49.7642567841484

x39 = 72.1044712096759

x40 = 20.0489132956248

x41 = 86.0671052256305

x42 = -47.974181327534

x43 = -95.8409490367987

x44 = 34.0115473115794

x45 = -35.8016227681938

x46 = 3.99188417727692

x47 = 46.2735982801598

x48 = -93.7465539344055

x49 = -15.8601230908384

x50 = -9.27261813787998

x51 = 42.0848080753734

x52 = -81.8783150208441

x53 = -37.896017870587

x54 = -79.7839199184509

x55 = 90.2558954304169

x56 = -42.3891277211522

x57 = 64.0312104458819

x58 = -7.87635473628452

x59 = 98.2396637849707

x60 = -21.8389887522392

x61 = -43.7853911227476

x62 = 74.198866312069

x63 = -39.9904129729802

x64 = -70.0100761072827

x65 = -100.029739241585

x66 = 47.974181327534

x67 = 14.1595400434641

x68 = -73.8050542570501

x69 = 70.0100761072827

x70 = -59.8424202410955

x71 = -61.9368153434887

x72 = 66.1256055482751

x73 = -26.0277789570255

x74 = -19.744593649846

x75 = 36.1059424139726

x76 = -1.89748907488373

x77 = -23.9333838546324

x78 = -1760.18692373859

x79 = 38.2003375163658

x80 = 82.1826346666229

x81 = 39.9904129729802

x82 = -83.9727101232373

x83 = 28.1221740594187

x84 = -34.0115473115794

x85 = 26.0277789570255

x86 = -27.7283620043998

x87 = 16.2539351458573

x88 = 100.029739241585

x89 = 96.1452686825775

x90 = 0.19690602750947

x91 = 50.0685764299272

x92 = 80.0882395642297

x93 = 8.18067438206331

x94 = 60.2362322961144

x95 = 54.2573666347136

x96 = 91.9564784777911

x97 = -75.8994493594433

x98 = 68.2200006506683

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