sin7x=1/2 (уравнение)

Найду корень уравнения: sin7x=1/2

Вы ввели

sin(7*x) = 1/2

$$\sin{\left(7 x \right)} = \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(7 x \right)} = \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$7 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$7 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Или
$$7 x = 2 \pi n + \frac{\pi}{6}$$
$$7 x = 2 \pi n + \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$7$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{7} + \frac{\pi}{42}$$
$$x_{2} = \frac{2 \pi n}{7} + \frac{5 \pi}{42}$$

Быстрый ответ

     pi
x1 = --
     42

$$x_{1} = \frac{\pi}{42}$$

     5*pi
x2 = ----
      42 

$$x_{2} = \frac{5 \pi}{42}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    42    42 

$$\left(0 + \frac{\pi}{42}\right) + \frac{5 \pi}{42}$$

=

pi
--
7 

$$\frac{\pi}{7}$$

произведение

  pi 5*pi
1*--*----
  42  42 

$$\frac{5 \pi}{42} \cdot 1 \frac{\pi}{42}$$

=

    2
5*pi 
-----
 1764

$$\frac{5 \pi^{2}}{1764}$$

Численный ответ

x1 = 26.1051389548295

x2 = -97.7633713867109

x3 = -67.8434413525224

x4 = -71.733032256967

x5 = -45.7026931272229

x6 = 90.1337892279929

x7 = 20.1211529479918

x8 = -31.9395253114962

x9 = -51.9858784344025

x10 = 82.055408118762

x11 = -8.00358128414543

x12 = 65.8986459003002

x13 = -95.9681755846596

x14 = -93.8737804822664

x15 = -22.065948400214

x16 = 83.8506039208133

x17 = -74.4258259600439

x18 = 29.994729859274

x19 = 3.96439072952998

x20 = 50.3402822825222

x21 = 78.1658172143175

x22 = -13.9875672909831

x23 = 68.2922403030353

x24 = 39.5691074702143

x25 = -75.9218224617533

x26 = -40.0179064207271

x27 = -37.3251127176502

x28 = 43.4586983746588

x29 = -225.222273332354

x30 = 73.9770270095311

x31 = 80.2602123167107

x32 = 28.1995340572227

x33 = -57.9698644412402

x34 = 21.9163487500431

x35 = 44.0570969753426

x36 = -33.7347211135475

x37 = 12.0427718387609

x38 = -4.11399037970092

x39 = -89.6849902774801

x40 = -15.7827630930344

x41 = 72.1818312074798

x42 = 76.669820712608

x43 = -101.952161591497

x44 = -1.72039597696584

x45 = -66.0482455504711

x46 = 16.2315620435473

x47 = 88.0393941255997

x48 = -79.8114133661978

x49 = 46.1514920777358

x50 = -5.90918618175223

x51 = 56.6234675897017

x52 = 52.1354780845735

x53 = 60.2138591938044

x54 = -41.8131022227784

x55 = -9.79877708619674

x56 = -27.7507351067098

x57 = 47.9466878797871

x58 = 86.2441983235484

x59 = 98.2121703372238

x60 = -48.096287529958

x61 = -69.9378364549156

x62 = -19.6723539974789

x63 = 64.1034500982489

x64 = 0.0747998250854713

x65 = -85.7953993730356

x66 = 42.2619011732913

x67 = 100.007366139275

x68 = -61.8594553456847

x69 = 62.0090549958557

x70 = 18.0267578455986

x71 = -35.8291162159407

x72 = -10.3971756868805

x73 = 24.3099431527782

x74 = -43.9074973251716

x75 = 8.15318093431637

x76 = 6.35798513226506

x77 = 96.1177752348306

x78 = 56.025068989018

x79 = -49.8914833320093

x80 = 34.1835200640604

x81 = -23.8611442022653

x82 = -80.7090112672235

x83 = -92.0785846802151

x84 = -87.8897944754287

x85 = -63.654651147736

x86 = 91.9289850300442

x87 = 2.16919492747867

x88 = -84.0002035709842

x89 = 12.9403697397865

x90 = -53.7810742364538

x91 = -59.7650602432915

x92 = 54.2298731869667

x93 = -25.9555393046585

x94 = -17.8771581954276

x95 = 94.3225794327793

x96 = 70.0874361050866

x97 = 38.0731109685049

x98 = 62.9066528968813

x99 = 10.2475760367096

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