sin(3x)=√2/2 (уравнение)

Найду корень уравнения: sin(3x)=√2/2

Вы ввели

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

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

Подробное решение

Дано уравнение
$$\sin{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
- это простейшее тригонометрическое ур-ние
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 = -41.6261026600648

x2 = 94.5095789954929

x3 = 50.5272818452358

x4 = -87.7027949127151

x5 = -29.0597320457056

x6 = -43.720497762458

x7 = -22.2529479629277

x8 = -70.4240353179712

x9 = 8.63937979737193

x10 = -96.0803753222878

x11 = -35.3429173528852

x12 = 32.2013246992954

x13 = -39.5317075576716

x14 = -62.0464549083984

x15 = 21.7293491873294

x16 = -68.329640215578

x17 = -89.7971900151083

x18 = 0.261799387799149

x19 = 74.0892267471593

x20 = 46.3384916404494

x21 = 4.45058959258554

x22 = 19.6349540849362

x23 = -55.7632696012188

x24 = 84.037603483527

x25 = 44.2440965380563

x26 = 90.3207887907066

x27 = -79.3252145031423

x28 = 104.981554507459

x29 = -24.3473430653209

x30 = -45.8148928648512

x31 = -26.4417381677141

x32 = -32.7249234748937

x33 = 78.2780169519457

x34 = -37.4373124552784

x35 = 52.621676947629

x36 = -8.11578102177363

x37 = -47.9092879672443

x38 = 28.012534494509

x39 = -1.83259571459405

x40 = -91.8915851175014

x41 = 23.8237442897226

x42 = 2.35619449019234

x43 = -66.2352451131848

x44 = -13.8753675533549

x45 = -18.0641577581413

x46 = 76.1836218495525

x47 = -97.6511716490827

x48 = -99.7455667514759

x49 = 17.540558982543

x50 = 6.54498469497874

x51 = -30.6305283725005

x52 = -57.857664703612

x53 = 48.4328867428426

x54 = 88.2263936883134

x55 = -85.6083998103219

x56 = -3.92699081698724

x57 = 81.9432083811338

x58 = 86.1319985859202

x59 = 92.4151838930998

x60 = 69.9004365423729

x61 = -83.5140047079287

x62 = 1127.04636447534

x63 = -9.68657734856853

x64 = 42.1497014356631

x65 = -15.9697626557481

x66 = 65.7116463375865

x67 = 30.1069295969022

x68 = 60.9992573572018

x69 = -93.9859802198946

x70 = 98.6983692002793

x71 = -73.0420291959627

x72 = -50.0036830696375

x73 = -11.7809724509617

x74 = -81.4196096055355

x75 = 71.9948316447661

x76 = -53.6688744988256

x77 = -52.0980781720307

x78 = 100.792764302673

x79 = -64.1408500107916

x80 = -273.580360250111

x81 = -20.1585528605345

x82 = -59.9520598060052

x83 = 37.9609112308767

x84 = 63.6172512351933

x85 = 44397.2491799188

x86 = 34.2957198016886

x87 = -76.7072206251508

x88 = 67.8060414399797

x89 = 25.9181393921158

x90 = 11.2573736753634

x91 = 40.0553063332699

x92 = 96.6039740978861

x93 = -1972.13478829099

x94 = -6.02138591938044

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