sin(6x)=1/2 (уравнение)

Найду корень уравнения: sin(6x)=1/2

Вы ввели

sin(6*x) = 1/2

$$\sin{\left(6 x \right)} = \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(6 x \right)} = \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$6 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$6 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Или
$$6 x = 2 \pi n + \frac{\pi}{6}$$
$$6 x = 2 \pi n + \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$6$$
получим ответ:
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{36}$$
$$x_{2} = \frac{\pi n}{3} + \frac{5 \pi}{36}$$

Быстрый ответ

     pi
x1 = --
     36

$$x_{1} = \frac{\pi}{36}$$

     5*pi
x2 = ----
      36 

$$x_{2} = \frac{5 \pi}{36}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    36    36 

$$\left(0 + \frac{\pi}{36}\right) + \frac{5 \pi}{36}$$

=

pi
--
6 

$$\frac{\pi}{6}$$

произведение

  pi 5*pi
1*--*----
  36  36 

$$\frac{5 \pi}{36} \cdot 1 \frac{\pi}{36}$$

=

    2
5*pi 
-----
 1296

$$\frac{5 \pi^{2}}{1296}$$

Численный ответ

x1 = -81.245076680336

x2 = 6.3704517697793

x3 = 16.1442955809475

x4 = -10.0356431989674

x5 = -27.8380015693096

x6 = -59.6029939556064

x7 = 11.9555053761612

x8 = -69.7259036171735

x9 = -89.9717229403077

x10 = -85.7829327355213

x11 = -61.6973890579996

x12 = 48.2583538176432

x13 = 60.1265927312047

x14 = -45.9894257900506

x15 = 44.0695636128568

x16 = -93.8114472946952

x17 = -91.717052192302

x18 = -2.00712863979348

x19 = 26.2672052425147

x20 = 9.86111027376796

x21 = 70.2495023927718

x22 = 96.7785070230856

x23 = 66.0607121879854

x24 = 88.0518607631139

x25 = -54.0179403492245

x26 = 4.27605666738611

x27 = 72.343897495165

x28 = -12.1300383013606

x29 = 80.02334620394

x30 = 40.2298392584693

x31 = 55.9378025264183

x32 = -71.8202987195667

x33 = -98.0002374994816

x34 = -39.706240482871

x35 = -49.8291501444381

x36 = -3.75245789178781

x37 = -34.1211868764891

x38 = 2.18166156499291

x39 = 33.9466539512897

x40 = 62.2209878335978

x41 = -37.6118453804778

x42 = -100.094632601875

x43 = -7.9412480965742

x44 = 68.1551072903786

x45 = 36.0410490536829

x46 = 24.1728101401215

x47 = 77.9289511015468

x48 = 84.2121364087264

x49 = 18.2386906833407

x50 = -41.8006355852642

x51 = -43.8950306876574

x52 = 31.8522588488965

x53 = -17.7150919077424

x54 = -87.8773278379145

x55 = -29.9323966717028

x56 = 22.0784150377283

x57 = -83.6885376331281

x58 = -78.1034840267462

x59 = -56.1123354516177

x60 = 38.1354441560761

x61 = 28.3616003449079

x62 = -63.7917841603927

x63 = -25.7436064669164

x64 = 92.2406509679003

x65 = -23.998277214922

x66 = -51.9235452468313

x67 = -32.0267917740959

x68 = -47.7347550420449

x69 = -19.8094870101356

x70 = 391.04101890933

x71 = -65.8861792627859

x72 = -21.9038821125288

x73 = 50.3527489200364

x74 = -15.6206968053492

x75 = 58.0321976288115

x76 = 53.4943415736262

x77 = 41.9751685104636

x78 = 75.4854901487547

x79 = 94.3350460702935

x80 = 99.9200996766754

x81 = -67.9805743651791

x82 = -5.846852994181

x83 = 82.1177413063332

x84 = -95.9058423970884

x85 = 46.16395871525

x86 = 19.9840199353351

x87 = 63.9663170855922

x88 = 62.9191195343956

x89 = 90.1462558655071

x90 = -73.9146938219599

x91 = -76.0090889243531

x92 = 0.0872664625997165

x93 = 14.0499004785544

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