2sin3x-√2=0 (уравнение)

Найду корень уравнения: 2sin3x-√2=0

Вы ввели

               ___    
2*sin(3*x) - \/ 2  = 0

$$2 \sin{\left(3 x \right)} - \sqrt{2} = 0$$

Подробное решение

Дано уравнение
$$2 \sin{\left(3 x \right)} - \sqrt{2} = 0$$
- это простейшее тригонометрическое ур-ние

Перенесём -sqrt(2) в правую часть ур-ния

с изменением знака при -sqrt(2)

Получим:
$$2 \sin{\left(3 x \right)} = \sqrt{2}$$

Разделим обе части ур-ния на 2

Ур-ние превратится в
$$\sin{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
Это ур-ние преобразуется в
$$3 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Или
$$3 x = 2 \pi n + \frac{\pi}{4}$$
$$3 x = 2 \pi n + \frac{3 \pi}{4}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$3$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{4}$$

Быстрый ответ

     pi
x1 = --
     12

$$x_{1} = \frac{\pi}{12}$$

     pi
x2 = --
     4 

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

Сумма и произведение корней

сумма

    pi   pi
0 + -- + --
    12   4 

$$\left(0 + \frac{\pi}{12}\right) + \frac{\pi}{4}$$

=

pi
--
3 

$$\frac{\pi}{3}$$

произведение

  pi pi
1*--*--
  12 4 

$$\frac{\pi}{4} \cdot 1 \frac{\pi}{12}$$

=

  2
pi 
---
 48

$$\frac{\pi^{2}}{48}$$

Численный ответ

x1 = 65.7116463375865

x2 = 28.012534494509

x3 = -39.5317075576716

x4 = 104.981554507459

x5 = -32.7249234748937

x6 = -11.7809724509617

x7 = -79.3252145031423

x8 = -3.92699081698724

x9 = 52.621676947629

x10 = -66.2352451131848

x11 = -15.9697626557481

x12 = -43.720497762458

x13 = -18.0641577581413

x14 = -73.0420291959627

x15 = -26.4417381677141

x16 = 50.5272818452358

x17 = -59.9520598060052

x18 = 63.6172512351933

x19 = 48.4328867428426

x20 = -76.7072206251508

x21 = -35.3429173528852

x22 = 71.9948316447661

x23 = -47.9092879672443

x24 = 92.4151838930998

x25 = 4.45058959258554

x26 = -50.0036830696375

x27 = 0.261799387799149

x28 = -97.6511716490827

x29 = 23.8237442897226

x30 = -52.0980781720307

x31 = 19.6349540849362

x32 = -68.329640215578

x33 = 60.9992573572018

x34 = 86.1319985859202

x35 = 42.1497014356631

x36 = -83.5140047079287

x37 = -96.0803753222878

x38 = 32.2013246992954

x39 = 40.0553063332699

x40 = 88.2263936883134

x41 = -91.8915851175014

x42 = -1.83259571459405

x43 = 100.792764302673

x44 = 67.8060414399797

x45 = -37.4373124552784

x46 = -41.6261026600648

x47 = -55.7632696012188

x48 = -20.1585528605345

x49 = 44.2440965380563

x50 = -6.02138591938044

x51 = 2.35619449019234

x52 = -85.6083998103219

x53 = -1972.13478829099

x54 = -81.4196096055355

x55 = 74.0892267471593

x56 = -53.6688744988256

x57 = 78.2780169519457

x58 = 90.3207887907066

x59 = -24.3473430653209

x60 = 11.2573736753634

x61 = 94.5095789954929

x62 = 46.3384916404494

x63 = -22.2529479629277

x64 = 34.2957198016886

x65 = 44397.2491799188

x66 = -70.4240353179712

x67 = 8.63937979737193

x68 = 96.6039740978861

x69 = 1127.04636447534

x70 = 30.1069295969022

x71 = 84.037603483527

x72 = 98.6983692002793

x73 = -57.857664703612

x74 = -273.580360250111

x75 = -64.1408500107916

x76 = 69.9004365423729

x77 = -30.6305283725005

x78 = -9.68657734856853

x79 = -93.9859802198946

x80 = -89.7971900151083

x81 = -8.11578102177363

x82 = -29.0597320457056

x83 = 17.540558982543

x84 = -45.8148928648512

x85 = 81.9432083811338

x86 = 37.9609112308767

x87 = 76.1836218495525

x88 = 25.9181393921158

x89 = 21.7293491873294

x90 = 6.54498469497874

x91 = -87.7027949127151

x92 = -99.7455667514759

x93 = -13.8753675533549

x94 = -62.0464549083984

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