sin2x=-1/8 (уравнение)

Найду корень уравнения: sin2x=-1/8

Вы ввели

sin(2*x) = -1/8

$$\sin{\left(2 x \right)} = - \frac{1}{8}$$

Подробное решение

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

Быстрый ответ

     pi   asin(1/8)
x1 = -- + ---------
     2        2    

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

     -asin(1/8) 
x2 = -----------
          2     

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

Сумма и произведение корней

сумма

    pi   asin(1/8)   asin(1/8)
0 + -- + --------- - ---------
    2        2           2    

$$- \frac{\operatorname{asin}{\left(\frac{1}{8} \right)}}{2} + \left(0 + \left(\frac{\operatorname{asin}{\left(\frac{1}{8} \right)}}{2} + \frac{\pi}{2}\right)\right)$$

=

pi
--
2 

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

произведение

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

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

=

-(pi + asin(1/8))*asin(1/8) 
----------------------------
             4              

$$- \frac{\left(\operatorname{asin}{\left(\frac{1}{8} \right)} + \pi\right) \operatorname{asin}{\left(\frac{1}{8} \right)}}{4}$$

Численный ответ

x1 = -50.3281463730207

x2 = -97.4520361768676

x3 = 91.04352303852

x4 = -39.2072442542884

x5 = -15.770627183533

x6 = 36.1909794318667

x7 = -36.0656516006986

x8 = -9.48744187635341

x9 = 94.1851156921098

x10 = 95.8812398500727

x11 = -72.3192949481493

x12 = -31.478590451482

x13 = -29.782466293519

x14 = 75.335559770571

x15 = 51.8989426998156

x16 = 73.8900912749442

x17 = 92.7396471964829

x18 = 50.2028185418527

x19 = -22.0538124907126

x20 = 59.627596502622

x21 = -73.7647634437761

x22 = 12.5037066987751

x23 = -89.4727267117251

x24 = 70.7484986213544

x25 = 56.4860038490322

x26 = 7.91664554955852

x27 = -45.490429561468

x28 = -75.4608876017391

x29 = 29.9077941246871

x30 = -54.9152075222373

x31 = 6.22052139159555

x32 = -86.3311340581353

x33 = -42.3488369078782

x34 = -1.50813241121086

x35 = 23.6246088175075

x36 = -67.4815781365965

x37 = -53.4697390266105

x38 = 81.6187450777506

x39 = -64.3399854830067

x40 = -0.0626639155840327

x41 = 1.63346024237893

x42 = -88.0272582160982

x43 = 26.7662014710973

x44 = -81.7440729089187

x45 = 14.1998308567381

x46 = 65.9107818098016

x47 = -102.039097326084

x48 = 80.1732765821238

x49 = -37.7617757586615

x50 = -7.79131771839045

x51 = -61.1983928294169

x52 = -23.4992809863394

x53 = -40.9033684122513

x54 = -80.0479487509557

x55 = 20.4830161639177

x56 = -6.34584922276362

x57 = -95.7559120189047

x58 = 86.4564618893033

x59 = 1694.95190052728

x60 = 45.615757392636

x61 = -66.0361096409697

x62 = 58.1821280069952

x63 = 34.4948552739037

x64 = 17734.2278655988

x65 = 64.4653133141748

x66 = 48.7573500462258

x67 = 37.6364479274935

x68 = -94.3104435232778

x69 = 42.4741647390462

x70 = -59.7529243337901

x71 = -14.07450302557

x72 = -20.3576883327496

x73 = 87.9019303849302

x74 = 100.468300999289

x75 = -51.7736148686476

x76 = 72.1939671169812

x77 = 28.2116699667241

x78 = 89.5980545428931

x79 = 15.6452993523649

x80 = 78.4771524241608

x81 = -44.0449610658411

x82 = -17.2160956791598

x83 = 4.77505289596872

x84 = -58.0568001758271

x85 = 43.9196332346731

x86 = -28.3369977978922

x87 = 67.6069059677646

x88 = 21.9284846595445

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