sin(x)=2/pi (уравнение)

Найду корень уравнения: sin(x)=2/pi

Вы ввели

         2 
sin(x) = --
         pi

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

Подробное решение

Дано уравнение
$$\sin{\left(x \right)} = \frac{2}{\pi}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{\pi} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{\pi} \right)} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{\pi} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{\pi} \right)} + \pi$$
, где n - любое целое число

Быстрый ответ

              /2 \
x1 = pi - asin|--|
              \pi/

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

         /2 \
x2 = asin|--|
         \pi/

$$x_{2} = \operatorname{asin}{\left(\frac{2}{\pi} \right)}$$

Сумма и произведение корней

сумма

             /2 \       /2 \
0 + pi - asin|--| + asin|--|
             \pi/       \pi/

$$\operatorname{asin}{\left(\frac{2}{\pi} \right)} + \left(0 + \left(\pi - \operatorname{asin}{\left(\frac{2}{\pi} \right)}\right)\right)$$

=

pi

$$\pi$$

произведение

  /         /2 \\     /2 \
1*|pi - asin|--||*asin|--|
  \         \pi//     \pi/

$$1 \left(\pi - \operatorname{asin}{\left(\frac{2}{\pi} \right)}\right) \operatorname{asin}{\left(\frac{2}{\pi} \right)}$$

=

/         /2 \\     /2 \
|pi - asin|--||*asin|--|
\         \pi//     \pi/

$$\left(\pi - \operatorname{asin}{\left(\frac{2}{\pi} \right)}\right) \operatorname{asin}{\left(\frac{2}{\pi} \right)}$$

Численный ответ

x1 = -85.513108738299

x2 = -55.8585606732417

x3 = -91.7962940454785

x4 = -22.6812556665031

x5 = 90.4160798627295

x6 = -10.1148850521439

x7 = 69.80514547035

x8 = 33.8674120981132

x9 = -18.1594488301642

x10 = -60.3803675095806

x11 = -43.2921900588826

x12 = 65.2833386340111

x13 = 50.9555895488112

x14 = 27.5842267909336

x15 = 15.0178561765744

x16 = -16.3980703593235

x17 = 19.5396630129133

x18 = -80.9913019019601

x19 = 25.8228483200929

x20 = -41.5308115880418

x21 = -79.2299234311194

x22 = 76.0883307775296

x23 = 88.6547013918888

x24 = 71.5665239411907

x25 = 82.3715160847092

x26 = 2.45148556221525

x27 = -72.9467381239398

x28 = -30.7258194445234

x29 = 21.301041483754

x30 = 371.39804021497

x31 = 52.7169680196519

x32 = 8.73467086939484

x33 = -11.8762635229846

x34 = -99.8408578234988

x35 = 0.69010709137454

x36 = -62.1417459804213

x37 = -47.8139968952214

x38 = 13.2564777057337

x39 = -93.5576725163193

x40 = -68.4249312876009

x41 = -123.212220581376

x42 = 84.1328945555499

x43 = 101.221072006248

x44 = -24.4426341373438

x45 = 63.5219601631704

x46 = 77.8497092483703

x47 = -3.83169974496433

x48 = 57.2387748559908

x49 = -98.0794793526581

x50 = -28.9644409736827

x51 = -66.6635528167602

x52 = -37.009004751703

x53 = 96.6992651699091

x54 = -49.5753753660622

x55 = 59.0001533268315

x56 = 40.1505974052928

x57 = 38.3892189344521

x58 = -35.2476262808623

x59 = -5.59307821580505

x60 = -54.097182202401

x61 = 46.4337827124724

x62 = 94.9378866990683

x63 = 22104.9360177492

x64 = 44.6724042416316

x65 = 6.97329239855413

x66 = -74.7081165947805

x67 = -475.070597783433

x68 = -225.504563967091

x69 = -87.2744872091397

x70 = 32.1060336272725

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