sin(4x)=-1 (уравнение)

Найду корень уравнения: sin(4x)=-1

Вы ввели

sin(4*x) = -1

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

Подробное решение

Дано уравнение
$$\sin{\left(4 x \right)} = -1$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Или
$$4 x = 2 \pi n - \frac{\pi}{2}$$
$$4 x = 2 \pi n + \frac{3 \pi}{2}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} - \frac{\pi}{8}$$
$$x_{2} = \frac{\pi n}{2} + \frac{3 \pi}{8}$$

Быстрый ответ

     -pi 
x1 = ----
      8  

$$x_{1} = - \frac{\pi}{8}$$

     3*pi
x2 = ----
      8  

$$x_{2} = \frac{3 \pi}{8}$$

Сумма и произведение корней

сумма

    pi   3*pi
0 - -- + ----
    8     8  

$$\left(- \frac{\pi}{8} + 0\right) + \frac{3 \pi}{8}$$

=

pi
--
4 

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

произведение

  -pi  3*pi
1*----*----
   8    8  

$$\frac{3 \pi}{8} \cdot 1 \left(- \frac{\pi}{8}\right)$$

=

     2
-3*pi 
------
  64  

$$- \frac{3 \pi^{2}}{64}$$

Численный ответ

x1 = -89.9280899395631

x2 = 12.1736716095332

x3 = -16.1006622734711

x4 = 21.5984495824462

x5 = -53.7997741489889

x6 = -88.3572932691129

x7 = 64.0099503040047

x8 = -14.5298663255924

x9 = -41.233403301565

x10 = 70.2931355951281

x11 = 87.5718952800362

x12 = -17.6714587255811

x13 = 57.7267651333619

x14 = 76.5763208145247

x15 = -97.7820713166882

x16 = 71.8639320431144

x17 = 42.0188017154946

x18 = 48.301987014692

x19 = -58.512163418571

x20 = -31.8086255639604

x21 = 86.0010988906895

x22 = -8.24668059419322

x23 = -30.237829170498

x24 = -74.2201263237444

x25 = -69.5077375714784

x26 = 32.594023666847

x27 = 98.5674693889903

x28 = -23.9546440698867

x29 = -63.2245518402327

x30 = 100.138265970845

x31 = 18.4568567377593

x32 = -66.3661446973152

x33 = -36.5210148980811

x34 = -52.2289777470169

x35 = -75.7909227331784

x36 = 43.5895981498367

x37 = 48.3019869823087

x38 = -0.392698986134672

x39 = 27.8816348852699

x40 = 34.1648201986395

x41 = 35.7356165577455

x42 = 65.5807467157997

x43 = 79.7179137087287

x44 = -82.0741080135087

x45 = 92.2842841757553

x46 = 56.1559687898059

x47 = 13.7444679818906

x48 = -19.2422547864247

x49 = -47.5165889944504

x50 = 46.7311907920314

x51 = -11.3882734788215

x52 = -3.53429183982371

x53 = -61.6537558849751

x54 = -1.96349547006081

x55 = 40.4480053155132

x56 = 78.1471173828293

x57 = -60.0829594332135

x58 = -99.3528677616324

x59 = 10.6028750937315

x60 = 84.4303027206186

x61 = 24.7400422576775

x62 = -77.3617191922663

x63 = 5.89048630601688

x64 = 62.4391538934323

x65 = -91.498886148304

x66 = -85.215700617134

x67 = -9.81747697788046

x68 = 93.8550806217218

x69 = -80.5033119347586

x70 = -38.0918108532085

x71 = -96.2112749006757

x72 = 68.7223393336083

x73 = 26.3108384344347

x74 = 4.31968985434627

x75 = -45.9457926779855

x76 = 20.0276531244896

x77 = -25.5254404172294

x78 = -39.6626073053534

x79 = -83.6449044644497

x80 = -44.374996126151

x81 = 2.74889373159412

x82 = -367.959039667966

x83 = 27.8816346273068

x84 = -33.3794220507366

x85 = 49.8727834642995

x86 = 84.4303024715116

x87 = -22.3838475557172

x88 = -67.9369412996327

x89 = 54.5851722404622

x90 = -55.370570621922

x91 = 90.7134878811712

График