sin(4x)=1/2 (уравнение)

Найду корень уравнения: sin(4x)=1/2

Вы ввели

sin(4*x) = 1/2

$$\sin{\left(4 x \right)} = \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(4 x \right)} = \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Или
$$4 x = 2 \pi n + \frac{\pi}{6}$$
$$4 x = 2 \pi n + \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{24}$$
$$x_{2} = \frac{\pi n}{2} + \frac{5 \pi}{24}$$

Быстрый ответ

     pi
x1 = --
     24

$$x_{1} = \frac{\pi}{24}$$

     5*pi
x2 = ----
      24 

$$x_{2} = \frac{5 \pi}{24}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    24    24 

$$\left(0 + \frac{\pi}{24}\right) + \frac{5 \pi}{24}$$

=

pi
--
4 

$$\frac{\pi}{4}$$

произведение

  pi 5*pi
1*--*----
  24  24 

$$\frac{5 \pi}{24} \cdot 1 \frac{\pi}{24}$$

=

    2
5*pi 
-----
 576 

$$\frac{5 \pi^{2}}{576}$$

Численный ответ

x1 = -59.0357619487082

x2 = -43.8513974563575

x3 = -37.0446133735796

x4 = 80.2415123604393

x5 = 60.3447588877039

x6 = -26.0490390860154

x7 = 47.7783882733448

x8 = 95.9494756283883

x9 = 73.9583270532597

x10 = 98.0438707307815

x11 = 0.130899693899575

x12 = -18.1950574520409

x13 = 51.9671784781312

x14 = -92.022484811401

x15 = -51.705379090332

x16 = 44.1131968441567

x17 = -76.8381203190504

x18 = 76.0527221556529

x19 = 83.9067037896274

x20 = 43.0659992929601

x21 = -59.5593607243065

x22 = -7.72308194007491

x23 = 2.22529479629277

x24 = 6.41408500107916

x25 = 94.3786793015934

x26 = -35.997415822383

x27 = -57.9885643975116

x28 = -79.9797129726402

x29 = 91.7606854236019

x30 = -84.1685031774265

x31 = 3.79609112308767

x32 = -21.860248881229

x33 = -55.8941692951184

x34 = 72.3875307264648

x35 = 17.9332580642417

x36 = 38.3536103125754

x37 = -0.916297857297023

x38 = -41.7570023539643

x39 = -11.9118721448613

x40 = -294.655210967943

x41 = 82.3359074628325

x42 = 58.2503637853107

x43 = 69.7695368484733

x44 = 92.8078829747985

x45 = -98.8292688941789

x46 = -70.0313362362725

x47 = -19.7658537788358

x48 = 86.5246976676189

x49 = -62.177354602298

x50 = 7.98488132787406

x51 = -73.6965276654606

x52 = 29.9760299030026

x53 = 39.9244066393703

x54 = -15.5770635740494

x55 = -77.885317870247

x56 = -21.3366501056307

x57 = -63.7481509290929

x58 = 68.1987405216784

x59 = 88.0954939944138

x60 = 14.2680666350536

x61 = -37.5682121491779

x62 = 54.0615735805244

x63 = 28.4052335762077

x64 = -85.7392995042214

x65 = -33.9030207199899

x66 = 32.0704250053958

x67 = -4.05789051088682

x68 = 22.1220482690281

x69 = -65.8425460314861

x70 = 46.2075919465499

x71 = -14.0062672472545

x72 = 16.3624617374468

x73 = 25.7872396982162

x74 = 36.2592152101822

x75 = 24.2164433714213

x76 = -29.7142305152035

x77 = -87.8336946066146

x78 = -73.1729288898623

x79 = -42.2806011295626

x80 = -95.6876762405891

x81 = -99.8764664453755

x82 = -81.550509299435

x83 = 90.189889096807

x84 = 50.3963821513363

x85 = 61.9155552144988

x86 = 74.481925828858

x87 = -40.1862060271694

x88 = 10.0792764302673

x89 = 66.1043454192852

x90 = -95.1640774649908

x91 = 21.0748507178315

x92 = -48.0401876611439

x93 = -1191.57991356783

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