sin(sin(x))=1/2 (уравнение)

Найду корень уравнения: sin(sin(x))=1/2

Вы ввели

sin(sin(x)) = 1/2

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

Подробное решение

Дано уравнение
$$\sin{\left(\sin{\left(x \right)} \right)} = \frac{1}{2}$$
преобразуем
$$\sin{\left(\sin{\left(x \right)} \right)} - \frac{1}{2} = 0$$
$$\sin{\left(\sin{\left(x \right)} \right)} - \frac{1}{2} = 0$$
Сделаем замену
$$w = \sin{\left(\sin{\left(x \right)} \right)}$$
Переносим свободные слагаемые (без w)
из левой части в правую, получим:
$$w = \frac{1}{2}$$
Получим ответ: w = 1/2
делаем обратную замену
$$\sin{\left(\sin{\left(x \right)} \right)} = w$$
подставляем w:

Быстрый ответ

              /pi\
x1 = pi - asin|--|
              \6 /

$$x_{1} = \pi - \operatorname{asin}{\left(\frac{\pi}{6} \right)}$$

         /pi\
x2 = asin|--|
         \6 /

$$x_{2} = \operatorname{asin}{\left(\frac{\pi}{6} \right)}$$

            /    /5*pi\\       /    /5*pi\\
x3 = pi - re|asin|----|| - I*im|asin|----||
            \    \ 6  //       \    \ 6  //

$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}$$

         /    /5*pi\\     /    /5*pi\\
x4 = I*im|asin|----|| + re|asin|----||
         \    \ 6  //     \    \ 6  //

$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}$$

Сумма и произведение корней

сумма

             /pi\       /pi\          /    /5*pi\\       /    /5*pi\\       /    /5*pi\\     /    /5*pi\\
0 + pi - asin|--| + asin|--| + pi - re|asin|----|| - I*im|asin|----|| + I*im|asin|----|| + re|asin|----||
             \6 /       \6 /          \    \ 6  //       \    \ 6  //       \    \ 6  //     \    \ 6  //

$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}\right) - \left(- 2 \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}\right)$$

=

2*pi

$$2 \pi$$

произведение

  /         /pi\\     /pi\ /       /    /5*pi\\       /    /5*pi\\\ /    /    /5*pi\\     /    /5*pi\\\
1*|pi - asin|--||*asin|--|*|pi - re|asin|----|| - I*im|asin|----|||*|I*im|asin|----|| + re|asin|----|||
  \         \6 //     \6 / \       \    \ 6  //       \    \ 6  /// \    \    \ 6  //     \    \ 6  ///

$$1 \left(\pi - \operatorname{asin}{\left(\frac{\pi}{6} \right)}\right) \operatorname{asin}{\left(\frac{\pi}{6} \right)} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}\right)$$

=

 /         /pi\\ /    /    /5*pi\\     /    /5*pi\\\ /          /    /5*pi\\     /    /5*pi\\\     /pi\
-|pi - asin|--||*|I*im|asin|----|| + re|asin|----|||*|-pi + I*im|asin|----|| + re|asin|----|||*asin|--|
 \         \6 // \    \    \ 6  //     \    \ 6  /// \          \    \ 6  //     \    \ 6  ///     \6 /

$$- \left(\pi - \operatorname{asin}{\left(\frac{\pi}{6} \right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{5 \pi}{6} \right)}\right)}\right) \operatorname{asin}{\left(\frac{\pi}{6} \right)}$$

Численный ответ

x1 = 57.0997373477157

x2 = -35.1085887725872

x3 = 34.0064496063883

x4 = 101.082034497973

x5 = 82.2324785764341

x6 = 2.59052307049035

x7 = -97.940441844383

x8 = 323.032973736649

x9 = 27.7232642992087

x10 = 31.9669961189974

x11 = -93.6967100245944

x12 = -85.3740712300239

x13 = 0.551069583099446

x14 = 96.8383026781841

x15 = -60.2413300013055

x16 = 59.1391908351066

x17 = 88.5156638836137

x18 = -72.8077006156647

x19 = -169.094933710749

x20 = -66.5245153084851

x21 = -49.7144128743372

x22 = -3.69266223668924

x23 = 25.6838108118178

x24 = -99.9798953317739

x25 = -12.0153010312597

x26 = 15.1568936848495

x27 = 38.250181426177

x28 = 6.83425489027903

x29 = -9.97584754386883

x30 = -79.0908859228443

x31 = -37.1480422599781

x32 = 52.856005527927

x33 = -55.9975981815168

x34 = 75.9492932692545

x35 = -5.73211572408014

x36 = -68.563968795876

x37 = 40.2896349135679

x38 = -18.2984863384393

x39 = 71.7055614494658

x40 = 50.8165520405361

x41 = 113.648405112332

x42 = -87.4135247174148

x43 = -688.559860719264

x44 = -43.4312275671577

x45 = 21.4400789920291

x46 = 63.3829226548953

x47 = -22.542218158228

x48 = -91.6572565372035

x49 = 94.7988491907932

x50 = -81.1303394102352

x51 = 84.271932063825

x52 = -47.6749593869463

x53 = 8.87370837766993

x54 = -160.772294916179

x55 = 77.9887467566454

x56 = 90.5551173710046

x57 = 69.6661079620749

x58 = -16.2590328510484

x59 = -30.8648569527985

x60 = -1155.55502693794

x61 = -74.8471541030556

x62 = -116.789997765922

x63 = -62.2807834886964

x64 = -41.3917740797668

x65 = 46.5728202207475

x66 = 44.5333667333566

x67 = -24.5816716456189

x68 = -53.9581446941259

x69 = 13.1174401974586

x70 = 19.4006255046382

x71 = 65.4223761422862

x72 = -28.8254034654076

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