sin(2x)=-1 (уравнение)

Найду корень уравнения: sin(2x)=-1

Вы ввели

sin(2*x) = -1

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

Подробное решение

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

Быстрый ответ

     -pi 
x1 = ----
      4  

$$x_{1} = - \frac{\pi}{4}$$

     3*pi
x2 = ----
      4  

$$x_{2} = \frac{3 \pi}{4}$$

Сумма и произведение корней

сумма

    pi   3*pi
0 - -- + ----
    4     4  

$$\left(- \frac{\pi}{4} + 0\right) + \frac{3 \pi}{4}$$

=

pi
--
2 

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

произведение

  -pi  3*pi
1*----*----
   4    4  

$$\frac{3 \pi}{4} \cdot 1 \left(- \frac{\pi}{4}\right)$$

=

     2
-3*pi 
------
  16  

$$- \frac{3 \pi^{2}}{16}$$

Численный ответ

x1 = -44.7676954314418

x2 = 49.4800845342934

x3 = 36.9137134309271

x4 = -82.4668069282144

x5 = 62.0464548195549

x6 = -19.6349541554673

x7 = -44.7676950635734

x8 = 27.4889359573966

x9 = 62.0464547256144

x10 = -85.608399894653

x11 = -7.06858355298869

x12 = 43.1968988259316

x13 = -38.4845103583538

x14 = -63.6172513149926

x15 = 2.35619442440536

x16 = 46.3384915843947

x17 = -91.8915845717239

x18 = 80.8960105951638

x19 = -95.0331777492301

x20 = -76.1836217239439

x21 = 87.1791959595656

x22 = -51.0508806461775

x23 = -22.7765468685351

x24 = -29.0597322955756

x25 = -88.7499922112266

x26 = 21.2057502602382

x27 = -0.785397916651749

x28 = -10.2101759860574

x29 = -60.4756583509459

x30 = 71.471233111112

x31 = 55.7632697006301

x32 = -38.4845097737495

x33 = -54.1924731445711

x34 = 30.6305281863893

x35 = -13.3517689698773

x36 = -95.0331780209134

x37 = 36.9137135281996

x38 = 93.462381687851

x39 = -3.92699107594367

x40 = 80.896010582516

x41 = -66.758843637313

x42 = 77.7544183301131

x43 = 65.1880478021829

x44 = -29.0597320982225

x45 = -73.0420291965934

x46 = -88.7499925537801

x47 = -98.1747703033985

x48 = 96.6039739212089

x49 = 55.7632697511726

x50 = 33.7721211721652

x51 = 68.3296401645238

x52 = -16.4933611966261

x53 = 52.6216767646073

x54 = -32.2013245652766

x55 = 14.9225648553046

x56 = 8.63937960822746

x57 = -51.0508808708243

x58 = 99.7455669089878

x59 = -60.4756585174615

x60 = -41.6261027352652

x61 = 90.3207887447523

x62 = 65.1880473923958

x63 = -7.0685837201977

x64 = 84.0376034199549

x65 = -76.1836225361187

x66 = 43.1968992294597

x67 = 58.9048620635881

x68 = -0.785398304611266

x69 = -35.3429178934093

x70 = 40.0553062099627

x71 = 58.9048620066655

x72 = -22.7765464900166

x73 = 11.7809725930891

x74 = 18.0641575837788

x75 = 5.49778738042301

x76 = 24.3473430043575

x77 = -57.3340661259123

x78 = -73.0420294459387

x79 = 21.2057506565108

x80 = -79.3252147038408

x81 = 96.6039739773235

x82 = -96953.4762811305

x83 = 87.1791963746667

x84 = -35.3429175479242

x85 = -66.7588439932285

x86 = 74.6128253428807

x87 = -101.316363786262

x88 = 14.9225649892472

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