sin(pi*sin(x))=-1 (уравнение)

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

    Решение

    Вы ввели [src]
    sin(pi*sin(x)) = -1
    sin(πsin(x))=1\sin{\left(\pi \sin{\left(x \right)} \right)} = -1
    Подробное решение
    Дано уравнение
    sin(πsin(x))=1\sin{\left(\pi \sin{\left(x \right)} \right)} = -1
    преобразуем
    sin(πsin(x))+1=0\sin{\left(\pi \sin{\left(x \right)} \right)} + 1 = 0
    sin(πsin(x))+1=0\sin{\left(\pi \sin{\left(x \right)} \right)} + 1 = 0
    Сделаем замену
    w=sin(πsin(x))w = \sin{\left(\pi \sin{\left(x \right)} \right)}
    Переносим свободные слагаемые (без w)
    из левой части в правую, получим:
    w=1w = -1
    Получим ответ: w = -1
    делаем обратную замену
    sin(πsin(x))=w\sin{\left(\pi \sin{\left(x \right)} \right)} = w
    подставляем w:
    График
    0-80-60-40-2020406080-1001002-2
    Быстрый ответ [src]
         -pi 
    x1 = ----
          6  
    x1=π6x_{1} = - \frac{\pi}{6}
         7*pi
    x2 = ----
          6  
    x2=7π6x_{2} = \frac{7 \pi}{6}
    x3 = pi - re(asin(3/2)) - I*im(asin(3/2))
    x3=re(asin(32))+πiim(asin(32))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))
    x4=re(asin(32))+iim(asin(32))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 [src]
    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                                                                           
    (re(asin(32))+iim(asin(32)))(2π+re(asin(32))+iim(asin(32)))\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
    2π2 \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                                                                           
    7π61(π6)(re(asin(32))+πiim(asin(32)))(re(asin(32))+iim(asin(32)))\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                                      
    7π2(re(asin(32))+iim(asin(32)))(π+re(asin(32))+iim(asin(32)))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 [src]
    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
    sin(pi*sin(x))=-1 (уравнение) /media/krcore-image-pods/hash/equation/f/f6/ecdab56f0d17ba2842067ef9b324e.png