sin5x=-1/2 (уравнение)

Найду корень уравнения: sin5x=-1/2

Вы ввели

sin(5*x) = -1/2

$$\sin{\left(5 x \right)} = - \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(5 x \right)} = - \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$5 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Или
$$5 x = 2 \pi n - \frac{\pi}{6}$$
$$5 x = 2 \pi n + \frac{7 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$5$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{5} - \frac{\pi}{30}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{7 \pi}{30}$$

Быстрый ответ

     -pi 
x1 = ----
      30 

$$x_{1} = - \frac{\pi}{30}$$

     7*pi
x2 = ----
      30 

$$x_{2} = \frac{7 \pi}{30}$$

Сумма и произведение корней

сумма

    pi   7*pi
0 - -- + ----
    30    30 

$$\left(- \frac{\pi}{30} + 0\right) + \frac{7 \pi}{30}$$

=

pi
--
5 

$$\frac{\pi}{5}$$

произведение

  -pi  7*pi
1*----*----
   30   30 

$$\frac{7 \pi}{30} \cdot 1 \left(- \frac{\pi}{30}\right)$$

=

     2
-7*pi 
------
 900  

$$- \frac{7 \pi^{2}}{900}$$

Численный ответ

x1 = -15.6032435128293

x2 = -54.1401133968641

x3 = -1.78023583703422

x4 = -85.9749189532407

x5 = 52.2551578047102

x6 = 84.0899633610868

x7 = 42.2020613132229

x8 = 70.2669556852917

x9 = 20.001473227855

x10 = 87.8598745453945

x11 = 63.9837703781121

x12 = 94.1430598525741

x13 = -31.5206462910176

x14 = -5.55014702134197

x15 = -10.157816246607

x16 = 82.4144472791722

x17 = 9.52949771588904

x18 = -89.7448301375484

x19 = -34.0339204138894

x20 = 76.1312619719927

x21 = -0.10471975511966

x22 = -99.7979266290358

x23 = -65.868725970266

x24 = -96.028015444728

x25 = 26.2846585350346

x26 = -27.7507351067098

x27 = -31.9395253114962

x28 = 53.9306738866248

x29 = -64.1932098883514

x30 = 77.8067780539072

x31 = 62.3082542961976

x32 = 18.3259571459405

x33 = 40.1076662108297

x34 = -23.9808239224021

x35 = -78.0162175641465

x36 = 50.9985207432743

x37 = 22.0958683302482

x38 = 45.9719724975306

x39 = -91.8392252399416

x40 = -91.0014671989843

x41 = -62.9365728269155

x42 = 38.4321501289151

x43 = -79.6917336460611

x44 = -69.6386371545737

x45 = -13.9277274309148

x46 = -67.9631210726592

x47 = 32.1489648217356

x48 = -52.0457182944709

x49 = -57.9100245811719

x50 = 6.17846555205993

x51 = -71.733032256967

x52 = -11.8333323285216

x53 = -59.5855406630864

x54 = 66.0781654805053

x55 = 86.18435846348

x56 = 33.8244809036501

x57 = -25.6563400043166

x58 = -81.7861287484543

x59 = 28.3790536374278

x60 = -8.0634211442138

x61 = -35.709436495804

x62 = -72.1519112774456

x63 = 60.2138591938044

x64 = -3.87463093942741

x65 = 96.2374549549673

x66 = 100.007366139275

x67 = 50.160762702317

x68 = 8.27286065445312

x69 = 74.0368668695995

x70 = 97.9129710368819

x71 = 92.8864227911382

x72 = 56.025068989018

x73 = 23.3525053916841

x74 = -21.8864288200089

x75 = 9.94837673636768

x76 = -47.8569280896845

x77 = 72.3613507876849

x78 = -45.7625329872913

x79 = -41.9926218029836

x80 = -49.1135651511204

x81 = 12.0427718387609

x82 = -37.8038315981972

x83 = -20.629791758573

x84 = -88.0693140556339

x85 = -55.8156294787787

x86 = -75.9218224617533

x87 = 1.98967534727354

x88 = -44.0870169053768

x89 = 30.0545697193424

x90 = 43.8775773951374

x91 = 16.2315620435473

x92 = -98.1224105471212

x93 = 89.9542696477877

x94 = -17.6976386152225

График