sin(2x)=-1/3 (уравнение)

Найду корень уравнения: sin(2x)=-1/3

Вы ввели

sin(2*x) = -1/3

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

Подробное решение

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

Быстрый ответ

     pi   asin(1/3)
x1 = -- + ---------
     2        2    

$$x_{1} = \frac{\operatorname{asin}{\left(\frac{1}{3} \right)}}{2} + \frac{\pi}{2}$$

     -asin(1/3) 
x2 = -----------
          2     

$$x_{2} = - \frac{\operatorname{asin}{\left(\frac{1}{3} \right)}}{2}$$

Сумма и произведение корней

сумма

    pi   asin(1/3)   asin(1/3)
0 + -- + --------- - ---------
    2        2           2    

$$- \frac{\operatorname{asin}{\left(\frac{1}{3} \right)}}{2} + \left(0 + \left(\frac{\operatorname{asin}{\left(\frac{1}{3} \right)}}{2} + \frac{\pi}{2}\right)\right)$$

=

pi
--
2 

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

произведение

  /pi   asin(1/3)\ -asin(1/3) 
1*|-- + ---------|*-----------
  \2        2    /      2     

$$1 \left(\frac{\operatorname{asin}{\left(\frac{1}{3} \right)}}{2} + \frac{\pi}{2}\right) \left(- \frac{\operatorname{asin}{\left(\frac{1}{3} \right)}}{2}\right)$$

=

-(pi + asin(1/3))*asin(1/3) 
----------------------------
             4              

$$- \frac{\left(\operatorname{asin}{\left(\frac{1}{3} \right)} + \pi\right) \operatorname{asin}{\left(\frac{1}{3} \right)}}{4}$$

Численный ответ

x1 = -57.9495456366841

x2 = 89.7053090820362

x3 = -15.877881722676

x4 = 14.3070853958811

x5 = 100.361046460146

x6 = 48.8646045853689

x7 = -23.3920264471964

x8 = 8.02390008870154

x9 = 42.5814192781893

x10 = 30.0150486638301

x11 = -1.40087787206784

x12 = -94.4176980624209

x13 = -0.169918454727061

x14 = -42.2415823687351

x15 = -97.5592907160107

x16 = 52.0061972389586

x17 = -9.59469641549644

x18 = -59.8601788729331

x19 = 70.8557531604974

x20 = 34.3876007347607

x21 = 95.9884943892158

x22 = -89.365472172582

x23 = 64.5725678533178

x24 = -13.967248486427

x25 = -50.4354009121638

x26 = -45.3831750223249

x27 = -7.68406317924742

x28 = -44.1522156049842

x29 = 59.520341963479

x30 = -67.3743235974535

x31 = 6548.81980486265

x32 = 26.8734560102403

x33 = 37.5291933883505

x34 = 92.846901735626

x35 = -88.1345127552413

x36 = -95.6486574797616

x37 = -20.2504337936066

x38 = -73.6575089046331

x39 = 12.3964521596321

x40 = -66.1433641801127

x41 = 78.3698978850178

x42 = 6.11326685245253

x43 = 4.88230743511175

x44 = -106.98406867678

x45 = 20.5902707030607

x46 = 68.9451199242484

x47 = 73.9973458140872

x48 = 65.8035272706586

x49 = 50.0955640027096

x50 = 67.7141605069076

x51 = 43.81237869553

x52 = 23.7318633566505

x53 = 28.1044154275811

x54 = 81.5114905386076

x55 = -79.9406942118127

x56 = -31.585844990625

x57 = -37.8690302978046

x58 = 21.8212301204015

x59 = -86.2238795189922

x60 = 86.5637164284464

x61 = -72.4265494872923

x62 = -75.5681421408821

x63 = -64.2327309438637

x64 = -53.5769935657535

x65 = 58.2893825461382

x66 = -51.6663603295045

x67 = 45.7230119317791

x68 = -6.45310376190665

x69 = -28.4442523370352

x70 = 15.5380448132219

x71 = 84.6530831921974

x72 = -35.9583970615556

x73 = -22.1610670298556

x74 = 94.0778611529667

x75 = 1.74071478152196

x76 = 36.2982339710097

x77 = 72.0867125778382

x78 = 61.430975199728

x79 = 56.3787493098892

x80 = 80.2805311212668

x81 = -81.8513274480617

x82 = 87.7946758457872

x83 = -428.65747876028

x84 = -29.675211754376

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