sin(cos(x))=(1/2) (уравнение)

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

Вы ввели

sin(cos(x)) = 1/2

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

Подробное решение

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

Быстрый ответ

           /pi\       
x1 = - acos|--| + 2*pi
           \6 /       

$$x_{1} = - \operatorname{acos}{\left(\frac{\pi}{6} \right)} + 2 \pi$$

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

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

                /    /5*pi\\
x3 = 2*pi - I*im|acos|----||
                \    \ 6  //

$$x_{3} = 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)}$$

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

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

Сумма и произведение корней

сумма

          /pi\              /pi\              /    /5*pi\\       /    /5*pi\\     /    /5*pi\\
0 + - acos|--| + 2*pi + acos|--| + 2*pi - I*im|acos|----|| + I*im|acos|----|| + re|acos|----||
          \6 /              \6 /              \    \ 6  //       \    \ 6  //     \    \ 6  //

$$\left(\left(\operatorname{acos}{\left(\frac{\pi}{6} \right)} + \left(0 + \left(- \operatorname{acos}{\left(\frac{\pi}{6} \right)} + 2 \pi\right)\right)\right) + \left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)}\right)$$

=

         /    /5*pi\\
4*pi + re|acos|----||
         \    \ 6  //

$$\operatorname{re}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)} + 4 \pi$$

произведение

  /      /pi\       \     /pi\ /           /    /5*pi\\\ /    /    /5*pi\\     /    /5*pi\\\
1*|- acos|--| + 2*pi|*acos|--|*|2*pi - I*im|acos|----|||*|I*im|acos|----|| + re|acos|----|||
  \      \6 /       /     \6 / \           \    \ 6  /// \    \    \ 6  //     \    \ 6  ///

$$1 \left(- \operatorname{acos}{\left(\frac{\pi}{6} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{\pi}{6} \right)} \left(2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{5 \pi}{6} \right)}\right)}\right)$$

=

/      /pi\       \ /           /    /5*pi\\\ /    /    /5*pi\\     /    /5*pi\\\     /pi\
|- acos|--| + 2*pi|*|2*pi - I*im|acos|----|||*|I*im|acos|----|| + re|acos|----|||*acos|--|
\      \6 /       / \           \    \ 6  /// \    \    \ 6  //     \    \ 6  ///     \6 /

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

Численный ответ

x1 = 17.8298291778433

x2 = 38.718838586773

x3 = 11.5466438706637

x4 = -49.2457557137412

x5 = 82.7011357370301

x6 = -80.6616822496392

x7 = -42.9625704065617

x8 = -70.1347651226709

x9 = -99.5112381711779

x10 = -24.1130144850229

x11 = 76.4179504298505

x12 = -1521.55057108116

x13 = 70.1347651226709

x14 = 13.5860973580546

x15 = -57.5683945083117

x16 = -93.2280528639983

x17 = -1.01972674369545

x18 = 63.8515798154913

x19 = -82.7011357370301

x20 = -19.8692826652342

x21 = -32.4356532795934

x22 = -68.09531163528

x23 = -74.3784969424596

x24 = 80.6616822496392

x25 = 99.5112381711779

x26 = -26.1524679724138

x27 = -264.913509645238

x28 = -55.5289410209208

x29 = -7.30291205087504

x30 = 55.5289410209208

x31 = -13.5860973580546

x32 = -36.6793850993821

x33 = -17.8298291778433

x34 = -38.718838586773

x35 = 93.2280528639983

x36 = 95.2675063513892

x37 = -63.8515798154913

x38 = 24.1130144850229

x39 = 49.2457557137412

x40 = 74.3784969424596

x41 = 57.5683945083117

x42 = 42.9625704065617

x43 = -86.9448675568188

x44 = -176.948915344724

x45 = -5.26345856348414

x46 = -95.2675063513892

x47 = 26.1524679724138

x48 = -457.652800680414

x49 = -30.3961997922025

x50 = 36.6793850993821

x51 = 5.26345856348414

x52 = 3781.45782817842

x53 = -45.0020238939526

x54 = 30.3961997922025

x55 = -88.9843210442097

x56 = 1.01972674369545

x57 = 7.30291205087504

x58 = -76.4179504298505

x59 = -51.2852092011321

x60 = 19.8692826652342

x61 = 88.9843210442097

x62 = 32.4356532795934

x63 = 68.09531163528

x64 = 86.9448675568188

x65 = -11.5466438706637

x66 = -61.8121263281004

x67 = 51.2852092011321

x68 = 61.8121263281004

x69 = 45.0020238939526

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