sin(5x)=1/2 (уравнение)

Найду корень уравнения: sin(5x)=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
$$5 x = 2 \pi n + \frac{\pi}{6}$$
$$5 x = 2 \pi n + \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$5$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{5} + \frac{\pi}{30}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{\pi}{6}$$

Быстрый ответ

     pi
x1 = --
     30

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

     pi
x2 = --
     6 

$$x_{2} = \frac{\pi}{6}$$

Сумма и произведение корней

сумма

    pi   pi
0 + -- + --
    30   6 

$$\left(0 + \frac{\pi}{30}\right) + \frac{\pi}{6}$$

=

pi
--
5 

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

произведение

  pi pi
1*--*--
  30 6 

$$\frac{\pi}{6} \cdot 1 \frac{\pi}{30}$$

=

  2
pi 
---
180

$$\frac{\pi^{2}}{180}$$

Численный ответ

x1 = 16.4410015537866

x2 = -13.7182879206754

x3 = 65.868725970266

x4 = -16.2315620435473

x5 = 41.9926218029836

x6 = -67.7536815624199

x7 = 84.2994028713261

x8 = -91.6297857297023

x9 = 18.1165176357011

x10 = -69.8480766648131

x11 = -25.865779514556

x12 = 92.2581042604203

x13 = -1.98967534727354

x14 = -49.7418836818384

x15 = -32.1489648217356

x16 = -5.75958653158129

x17 = -47.6474885794452

x18 = 54.1401133968641

x19 = -27.1224165759919

x20 = -57.7005850709325

x21 = -12.0427718387609

x22 = -79.9011731563004

x23 = 34.0339204138894

x24 = -20.001473227855

x25 = -43.8775773951374

x26 = -74.0368668695995

x27 = -23.7713844121628

x28 = 85.9749189532407

x29 = 98.1224105471212

x30 = 6.38790506229925

x31 = 1.36135681655558

x32 = -66.0781654805053

x33 = 31.9395253114962

x34 = -3.66519142918809

x35 = 74.2463063798388

x36 = 50.3702022125564

x37 = 38.2227106186758

x38 = -77.8067780539072

x39 = -58.5383431118898

x40 = 70.8952742160097

x41 = -37.5943920879579

x42 = 8.0634211442138

x43 = -81.576689238215

x44 = 82.2050077689329

x45 = -33.8244809036501

x46 = 12.6710903694788

x47 = 30.2640092295817

x48 = 64.1932098883514

x49 = -97.9129710368819

x50 = -63.9837703781121

x51 = 23.9808239224021

x52 = -59.7949801733257

x53 = 72.1519112774456

x54 = 43.2492588644195

x55 = -35.9188760060433

x56 = -71.104713726249

x57 = 48.2758071101632

x58 = 21.8864288200089

x59 = -86.18435846348

x60 = -87.8598745453945

x61 = -90.3731486682664

x62 = -53.9306738866248

x63 = 4.29350995990605

x64 = -56.025068989018

x65 = 52.0457182944709

x66 = 78.0162175641465

x67 = -30.0545697193424

x68 = 28.1696141271885

x69 = 75.9218224617533

x70 = -93.7241808320955

x71 = 55.8156294787787

x72 = -42.2020613132229

x73 = 99.7979266290358

x74 = 62.0988147859582

x75 = 83.4616448303688

x76 = 40.317105721069

x77 = 94.3524993628135

x78 = -98.7507290778392

x79 = 44.0870169053768

x80 = 22.7241868609662

x81 = -100.007366139275

x82 = 10.157816246607

x83 = 20.2109127380943

x84 = 0.10471975511966

x85 = -39.6887871903511

x86 = -9.94837673636768

x87 = -76.1312619719927

x88 = 11.8333323285216

x89 = -89.9542696477877

x90 = 96.028015444728

x91 = -45.9719724975306

x92 = -15.8126830230686

x93 = 60.4232987040437

x94 = 67.9631210726592

x95 = -22.0958683302482

x96 = 88.0693140556339

График