sin(8x+1)=pi/5 (уравнение)

Найду корень уравнения: sin(8x+1)=pi/5

Вы ввели

               pi
sin(8*x + 1) = --
               5 

$$\sin{\left(8 x + 1 \right)} = \frac{\pi}{5}$$

Подробное решение

Дано уравнение
$$\sin{\left(8 x + 1 \right)} = \frac{\pi}{5}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$8 x + 1 = 2 \pi n + \operatorname{asin}{\left(\frac{\pi}{5} \right)}$$
$$8 x + 1 = 2 \pi n - \operatorname{asin}{\left(\frac{\pi}{5} \right)} + \pi$$
Или
$$8 x + 1 = 2 \pi n + \operatorname{asin}{\left(\frac{\pi}{5} \right)}$$
$$8 x + 1 = 2 \pi n - \operatorname{asin}{\left(\frac{\pi}{5} \right)} + \pi$$
, где n - любое целое число
Перенесём
$$1$$
в правую часть ур-ния
с противоположным знаком, итого:
$$8 x = 2 \pi n - 1 + \operatorname{asin}{\left(\frac{\pi}{5} \right)}$$
$$8 x = 2 \pi n - 1 - \operatorname{asin}{\left(\frac{\pi}{5} \right)} + \pi$$
Разделим обе части полученного ур-ния на
$$8$$
получим ответ:
$$x_{1} = \frac{\pi n}{4} - \frac{1}{8} + \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8}$$
$$x_{2} = \frac{\pi n}{4} - \frac{1}{8} - \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8} + \frac{\pi}{8}$$

Быстрый ответ

               /pi\
           asin|--|
       1       \5 /
x1 = - - + --------
       8      8    

$$x_{1} = - \frac{1}{8} + \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8}$$

               /pi\     
           asin|--|     
       1       \5 /   pi
x2 = - - - -------- + --
       8      8       8 

$$x_{2} = - \frac{1}{8} - \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8} + \frac{\pi}{8}$$

Сумма и произведение корней

сумма

              /pi\             /pi\     
          asin|--|         asin|--|     
      1       \5 /     1       \5 /   pi
0 + - - + -------- + - - - -------- + --
      8      8         8      8       8 

$$\left(\left(- \frac{1}{8} + \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8}\right) + 0\right) - \left(- \frac{\pi}{8} + \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8} + \frac{1}{8}\right)$$

=

  1   pi
- - + --
  4   8 

$$- \frac{1}{4} + \frac{\pi}{8}$$

произведение

  /          /pi\\ /          /pi\     \
  |      asin|--|| |      asin|--|     |
  |  1       \5 /| |  1       \5 /   pi|
1*|- - + --------|*|- - - -------- + --|
  \  8      8    / \  8      8       8 /

$$1 \left(- \frac{1}{8} + \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8}\right) \left(- \frac{1}{8} - \frac{\operatorname{asin}{\left(\frac{\pi}{5} \right)}}{8} + \frac{\pi}{8}\right)$$

=

/        /pi\\ /             /pi\\
|1 - asin|--||*|1 - pi + asin|--||
\        \5 // \             \5 //
----------------------------------
                64                

$$\frac{\left(1 - \operatorname{asin}{\left(\frac{\pi}{5} \right)}\right) \left(- \pi + \operatorname{asin}{\left(\frac{\pi}{5} \right)} + 1\right)}{64}$$

Численный ответ

x1 = 55.7231933420627

x2 = -97.9919950838262

x3 = 2.31611823103621

x4 = 81.078786170792

x5 = -50.0827071165818

x6 = -84.0776797426831

x7 = 18.8094796623826

x8 = -54.0096979335691

x9 = 92.0743604583563

x10 = 68.2895639564219

x11 = -91.7088097766466

x12 = 28.234257623152

x13 = -83.8548281426721

x14 = 15.6678870087928

x15 = -85.425624469467

x16 = -61.8636795675436

x17 = -59.7303366773622

x18 = 88.1473696413691

x19 = -69.717661201518

x20 = 58.302239432266

x21 = 99.9283420923308

x22 = 66.1562210662405

x23 = 96.0013512753436

x24 = 52.0190541250864

x25 = 10.1700998650107

x26 = -99.7856430106321

x27 = -63.6573274943494

x28 = -37.7391881022336

x29 = -21.0229750708762

x30 = -39.872530992415

x31 = 30.0279055499579

x32 = -0.602622822542593

x33 = -72.0738556917104

x34 = 33.9548963669451

x35 = 86.5765733145742

x36 = -90.9234116132491

x37 = -6.88580812972218

x38 = -57.9366887505563

x39 = 40.8006282375112

x40 = 22.1739239159834

x41 = -21.8083732342737

x42 = -33.8121972852464

x43 = 54.1523970152678

x44 = 74.010202700215

x45 = -88.0046705596703

x46 = -11.8210487101179

x47 = -35.9455401754278

x48 = -7.67120629311963

x49 = -65.7906703845308

x50 = -51.8763550433877

x51 = 55.9460449420737

x52 = -29.8852064682592

x53 = 8.03675697482934

x54 = 72.2165547734091

x55 = 4.1097661578421

x56 = -10.027400783312

x57 = -81.7214852524908

x58 = 40.2380816741247

x59 = 44.165072491112

x60 = -47.7265126263895

x61 = -32.0185493584405

x62 = 77.9371935172022

x63 = -76.0008465086976

x64 = 46.2984153812933

x65 = 11.9637477918166

x66 = -43.7995218094023

x67 = -77.7944944355035

x68 = -73.8675036185163

x69 = -79.9278373256849

x70 = 83.9975272243708

x71 = 80.2933880073946

x72 = -19.6750303440923

x73 = 24.3072668061648

x74 = -17.8813824172865

x75 = 50.2254061982806

x76 = 98.1346941655249

x77 = -5704.38693701482

x78 = 48.0920633080992

x79 = 90.2807125315504

x80 = 70.0832118832278

x81 = -95.8586521936448

x82 = 36.3110908571375

x83 = -55.803345860375

x84 = 18.2469330989962

x85 = -3.74421547613239

x86 = -41.6661789192209

x87 = -28.0915585414533

x88 = 0.182775340854855

x89 = 94.2077033485377

x90 = -15.7480395271051

x91 = -13.9543916002992

x92 = 76.1435455903964

x93 = 6.24310904802346

x94 = -94.0650042668389

x95 = 64.5854247394456

x96 = 32.1612484401392

x97 = -25.7353640512609

x98 = 26.1009147329707

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