sin4x=-√2/2 (уравнение)

Найду корень уравнения: sin4x=-√2/2

Вы ввели

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

$$\sin{\left(4 x \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(4 x \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- 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
- -- + ----
  16    16 

$$- \frac{\pi}{16} + \frac{5 \pi}{16}$$

=

pi
--
4 

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

произведение

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

$$- \frac{\pi}{16} \frac{5 \pi}{16}$$

=

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

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

Численный ответ

x1 = -63.0282026126452

x2 = -19.4386045440868

x3 = 54.3888228152733

x4 = -47.712938426395

x5 = 21.7947990342792

x6 = -32.004975158446

x7 = 72.0602814917159

x8 = -96.0149254753381

x9 = -67.7405915930299

x10 = -55.5669200603695

x11 = -77.558068635498

x12 = 10.4065256650162

x13 = -61.8501053675491

x14 = -91.6952355766521

x15 = -39.8589567924205

x16 = 62.2428044492478

x17 = -8.05033117482385

x18 = 48.1056375080937

x19 = 84.6266521060751

x20 = 109.366694253095

x21 = 26.1144889329652

x22 = 87.7682447596649

x23 = 29.2560815865549

x24 = 98.3711199655304

x25 = -81.877758534184

x26 = 42.2151512826128

x27 = 20.2240027074843

x28 = 94.0514300668444

x29 = 197.723987635308

x30 = 18.2605072989907

x31 = 90.5171383315559

x32 = -59.8866099590554

x33 = -99.5492172106266

x34 = -90.1244392498572

x35 = -22.1874981159779

x36 = -89.7317401681585

x37 = 50.0691329165873

x38 = -37.8954613839269

x39 = -75.9872723087031

x40 = -23.7582944427728

x41 = 35.9319659754333

x42 = -0.196349540849362

x43 = 227.176418762712

x44 = -88.1609438413636

x45 = -66.169795266235

x46 = 99.9419162923253

x47 = -14.7262155637022

x48 = 7.2649330114264

x49 = 28.0779843414588

x50 = -1.76714586764426

x51 = -45.7494430179014

x52 = 70.0967860832223

x53 = -10.0138265833175

x54 = 43.7859476094077

x55 = 86.1974484328699

x56 = 76.3799713904018

x57 = -85.0193511877738

x58 = 84.2339530243763

x59 = 65.7770961845363

x60 = -80.3069622073891

x61 = -33.5757714852409

x62 = -36.7173641388307

x63 = 46.5348411812988

x64 = -44.1786466911065

x65 = 6.08683576633022

x66 = -25.7217898512664

x67 = -11.5846229101124

x68 = 79.9142631256904

x69 = -69.7040870015235

x70 = -58.3158136322605

x71 = -94.4441291485432

x72 = 57.9231145505618

x73 = 13.9408174003047

x74 = 33.9684705669396

x75 = -83.8412539426776

x76 = 32.3976742401447

x77 = -17.8678082172919

x78 = -3.73064127613788

x79 = 55.9596191420682

x80 = 51.2472301616835

x81 = -97.9784208838317

x82 = 4.1233403578366

x83 = 68.5259897564274

x84 = 11.9773219918111

x85 = 62.6355035309465

x86 = -15.9043128087983

x87 = -52.0326283250809

x88 = 77.9507677171967

x89 = -53.9961237335746

x90 = -30.0414797499524

x91 = 64.2062998577414

x92 = 24.5436926061703

x93 = -74.0237769002095

x94 = 2.55254403104171

x95 = 92.0879346583508

x96 = 40.2516558741192

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