2cos(3x)=-1 (уравнение)

Найду корень уравнения: 2cos(3x)=-1

Вы ввели

2*cos(3*x) = -1

$$2 \cos{\left(3 x \right)} = -1$$

Подробное решение

Дано уравнение
$$2 \cos{\left(3 x \right)} = -1$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на 2

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

Быстрый ответ

     2*pi
x1 = ----
      9  

$$x_{1} = \frac{2 \pi}{9}$$

     4*pi
x2 = ----
      9  

$$x_{2} = \frac{4 \pi}{9}$$

Сумма и произведение корней

сумма

    2*pi   4*pi
0 + ---- + ----
     9      9  

$$\left(0 + \frac{2 \pi}{9}\right) + \frac{4 \pi}{9}$$

=

2*pi
----
 3  

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

произведение

  2*pi 4*pi
1*----*----
   9    9  

$$\frac{4 \pi}{9} \cdot 1 \cdot \frac{2 \pi}{9}$$

=

    2
8*pi 
-----
  81 

$$\frac{8 \pi^{2}}{81}$$

Численный ответ

x1 = -62.1337213709981

x2 = 9.77384381116824

x3 = 13.9626340159546

x4 = -89.3608577021097

x5 = -97.7384381116825

x6 = 92.8515162060983

x7 = -55.8505360638185

x8 = -49.567350756639

x9 = -36.3028484414821

x10 = -65.6243798749868

x11 = -67.71877497738

x12 = 7.67944870877505

x13 = 66.3225115757845

x14 = -53.7561409614254

x15 = -25.8308729295161

x16 = 16.0570291183478

x17 = -71.9075651821664

x18 = 34.2084533390889

x19 = -18.151424220741

x20 = 45.3785605518526

x21 = -47.4729556542458

x22 = 11.8682389135614

x23 = 44.6804288510548

x24 = 71.9075651821664

x25 = 99.8328332140756

x26 = 60.0393262686049

x27 = -41.1897703470662

x28 = 78.190750489346

x29 = 27.9252680319093

x30 = 82.3795406941324

x31 = 40.4916386462684

x32 = -1.39626340159546

x33 = 6.98131700797732

x34 = 42.5860337486616

x35 = -57.9449311662117

x36 = 62.1337213709981

x37 = 95.6440430092893

x38 = -7.67944870877505

x39 = -11.8682389135614

x40 = -30.0196631343025

x41 = -27.9252680319093

x42 = -69.8131700797732

x43 = -91.4552528045029

x44 = -32.1140582366957

x45 = -3.49065850398866

x46 = 51.6617458590322

x47 = 80.2851455917392

x48 = 22.3402144255274

x49 = 55.8505360638185

x50 = -99.8328332140756

x51 = -23.7364778271229

x52 = 25.8308729295161

x53 = 64.2281164733913

x54 = 36.3028484414821

x55 = 48.8692190558412

x56 = -76.0963553869528

x57 = 74.0019602845596

x58 = -78.190750489346

x59 = 32.1140582366957

x60 = 86.5683308989187

x61 = -93.5496479068961

x62 = -21.6420827247297

x63 = 4.88692190558412

x64 = -38.3972435438752

x65 = -247.836753783195

x66 = 18.151424220741

x67 = -9.77384381116824

x68 = -51.6617458590322

x69 = 84.4739357965256

x70 = 1.39626340159546

x71 = 97.7384381116825

x72 = -95.6440430092893

x73 = 49.567350756639

x74 = 76.0963553869528

x75 = -80.2851455917392

x76 = 69.8131700797732

x77 = 87.2664625997165

x78 = 88.6627260013119

x79 = 24.4346095279206

x80 = 38.3972435438752

x81 = 68.4169066781777

x82 = -60.0393262686049

x83 = -34.2084533390889

x84 = -82.3795406941324

x85 = -45.3785605518526

x86 = 20.2458193231342

x87 = 57.9449311662117

x88 = 30.0196631343025

x89 = 53.7561409614254

x90 = -5.58505360638185

x91 = -16.0570291183478

x92 = -13.9626340159546

x93 = -74.0019602845596

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