sin4x=-(√2/2) (уравнение)

Найду корень уравнения: sin4x=-(√2/2)

Вы ввели

              ___ 
           -\/ 2  
sin(4*x) = -------
              2   

$$\sin{\left(4 x \right)} = - \frac{\sqrt{2}}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(4 x \right)} = - \frac{\sqrt{2}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
Или
$$4 x = 2 \pi n - \frac{\pi}{4}$$
$$4 x = 2 \pi n + \frac{5 \pi}{4}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} - \frac{\pi}{16}$$
$$x_{2} = \frac{\pi n}{2} + \frac{5 \pi}{16}$$

Быстрый ответ

     -pi 
x1 = ----
      16 

$$x_{1} = - \frac{\pi}{16}$$

     5*pi
x2 = ----
      16 

$$x_{2} = \frac{5 \pi}{16}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 - -- + ----
    16    16 

$$\left(- \frac{\pi}{16} + 0\right) + \frac{5 \pi}{16}$$

=

pi
--
4 

$$\frac{\pi}{4}$$

произведение

  -pi  5*pi
1*----*----
   16   16 

$$\frac{5 \pi}{16} \cdot 1 \left(- \frac{\pi}{16}\right)$$

=

     2
-5*pi 
------
 256  

$$- \frac{5 \pi^{2}}{256}$$

Численный ответ

x1 = 54.3888228152733

x2 = -45.7494430179014

x3 = -55.5669200603695

x4 = 13.9408174003047

x5 = -15.9043128087983

x6 = 35.9319659754333

x7 = -89.7317401681585

x8 = 109.366694253095

x9 = -90.1244392498572

x10 = -63.0282026126452

x11 = 46.5348411812988

x12 = -58.3158136322605

x13 = -61.8501053675491

x14 = -81.877758534184

x15 = -30.0414797499524

x16 = -83.8412539426776

x17 = -37.8954613839269

x18 = 55.9596191420682

x19 = -59.8866099590554

x20 = -25.7217898512664

x21 = 43.7859476094077

x22 = 4.1233403578366

x23 = -67.7405915930299

x24 = 77.9507677171967

x25 = 92.0879346583508

x26 = 72.0602814917159

x27 = -85.0193511877738

x28 = 10.4065256650162

x29 = 6.08683576633022

x30 = -66.169795266235

x31 = 64.2062998577414

x32 = 70.0967860832223

x33 = -8.05033117482385

x34 = 20.2240027074843

x35 = -97.9784208838317

x36 = 62.2428044492478

x37 = 24.5436926061703

x38 = -32.004975158446

x39 = 50.0691329165873

x40 = 33.9684705669396

x41 = -3.73064127613788

x42 = -94.4441291485432

x43 = -53.9961237335746

x44 = 65.7770961845363

x45 = -80.3069622073891

x46 = 197.723987635308

x47 = 84.2339530243763

x48 = -47.712938426395

x49 = -14.7262155637022

x50 = 32.3976742401447

x51 = 86.1974484328699

x52 = -11.5846229101124

x53 = -91.6952355766521

x54 = 11.9773219918111

x55 = 79.9142631256904

x56 = 18.2605072989907

x57 = 76.3799713904018

x58 = 51.2472301616835

x59 = 28.0779843414588

x60 = 90.5171383315559

x61 = -99.5492172106266

x62 = 84.6266521060751

x63 = -1.76714586764426

x64 = -75.9872723087031

x65 = -22.1874981159779

x66 = -88.1609438413636

x67 = 68.5259897564274

x68 = 57.9231145505618

x69 = 48.1056375080937

x70 = -17.8678082172919

x71 = -33.5757714852409

x72 = -10.0138265833175

x73 = -52.0326283250809

x74 = -44.1786466911065

x75 = -77.558068635498

x76 = -0.196349540849362

x77 = 40.2516558741192

x78 = 21.7947990342792

x79 = -19.4386045440868

x80 = 87.7682447596649

x81 = 42.2151512826128

x82 = -74.0237769002095

x83 = 227.176418762712

x84 = -69.7040870015235

x85 = 62.6355035309465

x86 = 98.3711199655304

x87 = 99.9419162923253

x88 = -96.0149254753381

x89 = 29.2560815865549

x90 = 2.55254403104171

x91 = -36.7173641388307

x92 = 26.1144889329652

x93 = -39.8589567924205

x94 = -23.7582944427728

x95 = 94.0514300668444

x96 = 7.2649330114264

График