sin(-8x)=2/9 (уравнение)

Найду корень уравнения: sin(-8x)=2/9

Вы ввели

sin(-8*x) = 2/9

$$\sin{\left(- 8 x \right)} = \frac{2}{9}$$

Подробное решение

Дано уравнение
$$\sin{\left(- 8 x \right)} = \frac{2}{9}$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на -1

Ур-ние превратится в
$$\sin{\left(8 x \right)} = - \frac{2}{9}$$
Это ур-ние преобразуется в
$$8 x = 2 \pi n + \operatorname{asin}{\left(- \frac{2}{9} \right)}$$
$$8 x = 2 \pi n - \operatorname{asin}{\left(- \frac{2}{9} \right)} + \pi$$
Или
$$8 x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{9} \right)}$$
$$8 x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{9} \right)} + \pi$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$8$$
получим ответ:
$$x_{1} = \frac{\pi n}{4} - \frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8}$$
$$x_{2} = \frac{\pi n}{4} + \frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8} + \frac{\pi}{8}$$

Быстрый ответ

     pi   asin(2/9)
x1 = -- + ---------
     8        8    

$$x_{1} = \frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8} + \frac{\pi}{8}$$

     -asin(2/9) 
x2 = -----------
          8     

$$x_{2} = - \frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8}$$

Сумма и произведение корней

сумма

    pi   asin(2/9)   asin(2/9)
0 + -- + --------- - ---------
    8        8           8    

$$- \frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8} + \left(0 + \left(\frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8} + \frac{\pi}{8}\right)\right)$$

=

pi
--
8 

$$\frac{\pi}{8}$$

произведение

  /pi   asin(2/9)\ -asin(2/9) 
1*|-- + ---------|*-----------
  \8        8    /      8     

$$1 \left(\frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8} + \frac{\pi}{8}\right) \left(- \frac{\operatorname{asin}{\left(\frac{2}{9} \right)}}{8}\right)$$

=

-(pi + asin(2/9))*asin(2/9) 
----------------------------
             64             

$$- \frac{\left(\operatorname{asin}{\left(\frac{2}{9} \right)} + \pi\right) \operatorname{asin}{\left(\frac{2}{9} \right)}}{64}$$

Численный ответ

x1 = 45.9738041952884

x2 = 98.1467587881434

x3 = -31.780613981059

x4 = -5.86247458894319

x5 = 60.1109711364425

x6 = 84.0095918469893

x7 = 92.3122958357378

x8 = -95.8465875710264

x9 = 86.0291105285583

x10 = 28.2463222457705

x11 = -69.9284481789106

x12 = -35.7076047980462

x13 = -89.9000780724702

x14 = -29.8731418456407

x15 = 94.2197679711561

x16 = -0.0280116365376714

x17 = -67.9089294973416

x18 = -47.937299603782

x19 = -27.8536231640717

x20 = 21.9631369385909

x21 = -89.5634022638468

x22 = 43.9542855137194

x23 = -73.8554389958978

x24 = -85.9730872554829

x25 = -19.2142433666998

x26 = 65.945434088848

x27 = 34.1928317443267

x28 = 23.9826556201598

x29 = 64.0379619534297

x30 = -37.7271234796152

x31 = 56.1839803194552

x32 = -66.0014573619233

x33 = -91.9195967540391

x34 = -45.9177809222131

x35 = -62.4111423535595

x36 = -19.9996415300973

x37 = -77.7824298128851

x38 = -93.8270688894574

x39 = 52.256989502468

x40 = 10.1821644876292

x41 = 76.1556102130148

x42 = -25.9461510286535

x43 = -51.8642904207693

x44 = -79.6899019483033

x45 = -57.6987533731748

x46 = -53.7717625561875

x47 = -11.8089840874994

x48 = -3.95500245352491

x49 = -84.0656151200646

x50 = -87.9926059370519

x51 = 8.27469235221088

x52 = -9.78946540593043

x53 = 78.1751288945838

x54 = -13.7164562229177

x55 = 68.3016285790403

x56 = -71.8359203143288

x57 = 70.6578230692327

x58 = -7.88199327051215

x59 = -33.800132662628

x60 = -15.7359749044866

x61 = -23.9266323470845

x62 = 42.0468133783012

x63 = 100.166277469712

x64 = 14.1091553046164

x65 = 1.99150704503129

x66 = 87.9365826639765

x67 = -41.9907901052258

x68 = -44.3469845954182

x69 = 40.0272946967322

x70 = -49.8447717392003

x71 = 74.2481380775965

x72 = 62.0184432718607

x73 = -97.7540597064447

x74 = 27.460924082373

x75 = -75.7629111313161

x76 = -63.9819386803544

x77 = 38.1198225613139

x78 = 18.0361461216036

x79 = 6.25517367064192

x80 = -22.0191602116662

x81 = 54.1644616378863

x82 = 30.2658409273394

x83 = 32.1733130627577

x84 = -1.93548377195595

x85 = 20.0556648031726

x86 = -55.7912812377565

x87 = -99.7735783880136

x88 = -44.0103087867948

x89 = 80.0826010300021

x90 = 16.1286739861854

x91 = 58.0914524548735

x92 = 47.5446005220833

x93 = 96.2392866527251

x94 = 82.102119711571

x95 = 50.237470820899

x96 = 93.4343698077587

x97 = 36.1003038797449

x98 = 12.2016831691981

x99 = 72.2286193960276

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