2sin3x=-1 (уравнение)

Найду корень уравнения: 2sin3x=-1

Вы ввели

2*sin(3*x) = -1

$$2 \sin{\left(3 x \right)} = -1$$

Подробное решение

Дано уравнение
$$2 \sin{\left(3 x \right)} = -1$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на 2

Ур-ние превратится в
$$\sin{\left(3 x \right)} = - \frac{1}{2}$$
Это ур-ние преобразуется в
$$3 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Или
$$3 x = 2 \pi n - \frac{\pi}{6}$$
$$3 x = 2 \pi n + \frac{7 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$3$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{18}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{7 \pi}{18}$$

Быстрый ответ

     -pi 
x1 = ----
      18 

$$x_{1} = - \frac{\pi}{18}$$

     7*pi
x2 = ----
      18 

$$x_{2} = \frac{7 \pi}{18}$$

Сумма и произведение корней

сумма

    pi   7*pi
0 - -- + ----
    18    18 

$$\left(- \frac{\pi}{18} + 0\right) + \frac{7 \pi}{18}$$

=

pi
--
3 

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

произведение

  -pi  7*pi
1*----*----
   18   18 

$$\frac{7 \pi}{18} \cdot 1 \left(- \frac{\pi}{18}\right)$$

=

     2
-7*pi 
------
 324  

$$- \frac{7 \pi^{2}}{324}$$

Численный ответ

x1 = 28.4488668075076

x2 = -63.704517697793

x3 = -61.6101225953998

x4 = -83.9503370209273

x5 = -16.929693744345

x6 = -48.3456202802429

x7 = 43.8077642250577

x8 = 87.7900613753148

x9 = 20.0712863979348

x10 = -90.2335223281068

x11 = 39.6189740202713

x12 = 96.1676417848876

x13 = -65.7989128001862

x14 = -2.26892802759263

x15 = -23.9110107523223

x16 = -44.1568300754565

x17 = -21.8166156499291

x18 = 15.8824961931484

x19 = -31.5904594610974

x20 = -39.9680398706701

x21 = 6.10865238198015

x22 = 30.5432619099008

x23 = -98.6111027376796

x24 = -17.6278254451427

x25 = -15.5334303427495

x26 = 70.3367688553715

x27 = 76.6199541625511

x28 = -74.176493209759

x29 = -67.8933079025794

x30 = 8.20304748437335

x31 = -7.15584993317675

x32 = 83.6012711705284

x33 = 4.01425727958696

x34 = -26.0054058547155

x35 = 66.1479786505851

x36 = 22.165681500328

x37 = 77.3180858633488

x38 = 56.3741348394168

x39 = 12.3918376891597

x40 = 45.9021593274509

x41 = 32.637657012294

x42 = 33.3357887130917

x43 = -35.7792496658838

x44 = -92.3279174305

x45 = -69.9877030049726

x46 = -86.0447321233204

x47 = 24.2600766027212

x48 = -77.6671517137477

x49 = -42.0624349730633

x50 = 41.7133691226645

x51 = -4.36332312998582

x52 = -57.4213323906134

x53 = -75.5727566113545

x54 = 72.4311639577647

x55 = -51.1381470834339

x56 = -28.0998009571087

x57 = -81.8559419185341

x58 = 94.0732466824944

x59 = -37.873644768277

x60 = 646.2954220135

x61 = 89.884456477708

x62 = 64.0535835481919

x63 = 47.9965544298441

x64 = -59.5157274930066

x65 = -11.3446401379631

x66 = 54.2797397370236

x67 = 74.5255590601579

x68 = -19.7222205475359

x69 = -72.0820981073658

x70 = 98.2620368872808

x71 = 61.9591884457987

x72 = -0.174532925199433

x73 = 10.2974425867665

x74 = 68.2423737529783

x75 = 1.91986217719376

x76 = -79.7615468161409

x77 = -13.4390352403563

x78 = -33.6848545634906

x79 = 50.0909495322373

x80 = 91.9788515801012

x81 = 59.8647933434055

x82 = 85.6956662729216

x83 = 26.3544717051144

x84 = -46.2512251778497

x85 = -88.1391272257137

x86 = 100.356431989674

x87 = 52.1853446346305

x88 = 17.9768912955416

x89 = -54.6288055874225

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