sin3x+1=0 (уравнение)

Найду корень уравнения: sin3x+1=0

Вы ввели

sin(3*x) + 1 = 0

$$\sin{\left(3 x \right)} + 1 = 0$$

Подробное решение

Дано уравнение
$$\sin{\left(3 x \right)} + 1 = 0$$
- это простейшее тригонометрическое ур-ние

Перенесём 1 в правую часть ур-ния

с изменением знака при 1

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

Быстрый ответ

     -pi 
x1 = ----
      6  

$$x_{1} = - \frac{\pi}{6}$$

     pi
x2 = --
     2 

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

Сумма и произведение корней

сумма

    pi   pi
0 - -- + --
    6    2 

$$\left(- \frac{\pi}{6} + 0\right) + \frac{\pi}{2}$$

=

pi
--
3 

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

произведение

  -pi  pi
1*----*--
   6   2 

$$\frac{\pi}{2} \cdot 1 \left(- \frac{\pi}{6}\right)$$

=

   2 
-pi  
-----
  12 

$$- \frac{\pi^{2}}{12}$$

Численный ответ

x1 = 14.1371670949785

x2 = 85.3466004238277

x3 = 41.3643033140264

x4 = 9.94837678401611

x5 = 12.0427717609223

x6 = -23.5619450047453

x7 = -25.6563400624587

x8 = -84.2994027162129

x9 = -88.4881929146437

x10 = -29.8451300978728

x11 = -2.61799380552474

x12 = 28.7979325198376

x13 = -69.6386372219478

x14 = -80.1106125819736

x15 = -46.6002909322334

x16 = -42.4115006545154

x17 = -44.5058957690364

x18 = -27.7507351061628

x19 = 45.5530936447347

x20 = -78.0162175465002

x21 = 68.5914394517514

x22 = 93.724180981475

x23 = 32.9867232717324

x24 = -36.1283154225025

x25 = -38.2227104853721

x26 = -59.1666617558465

x27 = 51.8362788945866

x28 = -57.0722664631985

x29 = -65.449847091496

x30 = -82.2050076419441

x31 = 7.85398173632972

x32 = 83.2522049224581

x33 = -6.80678413492343

x34 = -10.9955742377112

x35 = -92.676983354992

x36 = 1.57079649551977

x37 = 5.7595866707223

x38 = -48.6946862836254

x39 = 16.2315620579746

x40 = 62.3082542579151

x41 = -19.3731548549645

x42 = -15.1843646196319

x43 = 3.66519159119344

x44 = -71.733032269064

x45 = -96.8657738808645

x46 = -17.2787597552673

x47 = 56.0250689520163

x48 = 70.6858345434363

x49 = 22.5147472195373

x50 = 76.9690203132814

x51 = 24.6091422981059

x52 = 95.8185760523108

x53 = -90.5825880642629

x54 = 87.4409956573616

x55 = 26.7035373935834

x56 = -40.3171055622449

x57 = 74.8746248486494

x58 = 60.213859223833

x59 = 20.4203521530692

x60 = -13.0899693729028

x61 = -34.0339203801376

x62 = -42.4115010822267

x63 = 53.9306739466035

x64 = -61.2610569029623

x65 = 39.2699077976373

x66 = 97.9129711080124

x67 = 18.3259570974782

x68 = 72.7802296618859

x69 = 30.8923277323824

x70 = -4.71238923573352

x71 = 539.830337690668

x72 = -86.3937978052948

x73 = -67.544242162189

x74 = 91.6297858963776

x75 = -21.467549936198

x76 = 47.6474887440776

x77 = 64.4026493112537

x78 = -31.9395254618874

x79 = 43.4586985173418

x80 = 89.5353907930327

x81 = -0.523598624935135

x82 = 100.007366129944

x83 = 49.7418838263396

x84 = -75.9218227439853

x85 = -63.3554520074103

x86 = -73.8274272803725

x87 = 66.4970443756899

График