sin(pi*sin(x))=-1 (уравнение)

Найду корень уравнения: sin(pi*sin(x))=-1

Вы ввели

sin(pi*sin(x)) = -1

$$\sin{\left(\pi \sin{\left(x \right)} \right)} = -1$$

Подробное решение

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

Быстрый ответ

     -pi 
x1 = ----
      6  

$$x_{1} = - \frac{\pi}{6}$$

     7*pi
x2 = ----
      6  

$$x_{2} = \frac{7 \pi}{6}$$

x3 = pi - re(asin(3/2)) - I*im(asin(3/2))

$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}$$

x4 = I*im(asin(3/2)) + re(asin(3/2))

$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}$$

Сумма и произведение корней

сумма

    pi   7*pi                                                                         
0 - -- + ---- + pi - re(asin(3/2)) - I*im(asin(3/2)) + I*im(asin(3/2)) + re(asin(3/2))
    6     6                                                                           

$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) - \left(- 2 \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right)$$

=

2*pi

$$2 \pi$$

произведение

  -pi  7*pi                                                                         
1*----*----*(pi - re(asin(3/2)) - I*im(asin(3/2)))*(I*im(asin(3/2)) + re(asin(3/2)))
   6    6                                                                           

$$\frac{7 \pi}{6} \cdot 1 \left(- \frac{\pi}{6}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right)$$

=

    2                                                                          
7*pi *(I*im(asin(3/2)) + re(asin(3/2)))*(-pi + I*im(asin(3/2)) + re(asin(3/2)))
-------------------------------------------------------------------------------
                                       36                                      

$$\frac{7 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\frac{3}{2} \right)}\right)}\right)}{36}$$

Численный ответ

x1 = -25.6563400575784

x2 = 87.4409956755458

x3 = 5.75958669198159

x4 = -65.4498471137318

x5 = 56.0250689535476

x6 = -15.184364636283

x7 = -34.0339203817864

x8 = -31.9395254382276

x9 = 60.2138592227626

x10 = -1133.59134884904

x11 = -2.61799415622207

x12 = 22.5147472003714

x13 = -19.3731547998892

x14 = 72.7802296386064

x15 = 97.9129711005109

x16 = 30.8923281905739

x17 = -6.8067837124283

x18 = 66.4970443581517

x19 = 68.5914394977552

x20 = -44.5058957424715

x21 = 41.3643028836498

x22 = 43.4586985390256

x23 = -46.6002912131247

x24 = -59.1666617677488

x25 = 74.8746252321522

x26 = -0.523598600500097

x27 = -21.467549956936

x28 = 79.063415174746

x29 = -71.73303245649

x30 = -90.5825882917856

x31 = 12.0427717675619

x32 = 18.3259571008295

x33 = -78.0162175469491

x34 = 100.007366130042

x35 = 24.6091423484871

x36 = -38.2227104657936

x37 = -132.470489694285

x38 = -88.4881928864687

x39 = 28.7979324994445

x40 = -71.7330322692361

x41 = 91.629785835609

x42 = 49.7418838491948

x43 = 85.3466004511484

x44 = 9.94837678070913

x45 = -13.0899690526506

x46 = -69.6386372153118

x47 = -40.3171056108353

x48 = 62.3082542599839

x49 = 3.66519153871422

x50 = -84.2994027610386

x51 = -63.3554519472469

x52 = 47.6474886875546

x53 = -78.0162175295362

x54 = 93.7241810058253

x55 = 16.2315620577281

x56 = -82.2050076241151

x57 = 816648.109121656

x58 = 53.9306739413208

x59 = -75.9218226841762

x60 = -27.7507351061631

x61 = 85.3465999673719

График