sin(sin(sin(x)))=0 (уравнение)

Найду корень уравнения: sin(sin(sin(x)))=0

Вы ввели

sin(sin(sin(x))) = 0

$$\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} = 0$$

Подробное решение

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

Быстрый ответ

x1 = 0

$$x_{1} = 0$$

x2 = pi

$$x_{2} = \pi$$

x3 = pi - re(asin(pi)) - I*im(asin(pi))

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

x4 = pi - re(asin(pi - asin(pi))) - I*im(asin(pi - asin(pi)))

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

x5 = pi - re(asin(asin(pi))) - I*im(asin(asin(pi)))

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

x6 = I*im(asin(pi)) + re(asin(pi))

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

x7 = I*im(asin(pi - asin(pi))) + re(asin(pi - asin(pi)))

$$x_{7} = \operatorname{re}{\left(\operatorname{asin}{\left(\pi - \operatorname{asin}{\left(\pi \right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\pi - \operatorname{asin}{\left(\pi \right)} \right)}\right)}$$

x8 = I*im(asin(asin(pi))) + re(asin(asin(pi)))

$$x_{8} = \operatorname{re}{\left(\operatorname{asin}{\left(\operatorname{asin}{\left(\pi \right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\operatorname{asin}{\left(\pi \right)} \right)}\right)}$$

Сумма и произведение корней

сумма

pi + pi - re(asin(pi)) - I*im(asin(pi)) + pi - re(asin(pi - asin(pi))) - I*im(asin(pi - asin(pi))) + pi - re(asin(asin(pi))) - I*im(asin(asin(pi))) + I*im(asin(pi)) + re(asin(pi)) + I*im(asin(pi - asin(pi))) + re(asin(pi - asin(pi))) + I*im(asin(asin(pi))) + re(asin(asin(pi)))

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

=

4*pi

$$4 \pi$$

произведение

0*pi*(pi - re(asin(pi)) - I*im(asin(pi)))*(pi - re(asin(pi - asin(pi))) - I*im(asin(pi - asin(pi))))*(pi - re(asin(asin(pi))) - I*im(asin(asin(pi))))*(I*im(asin(pi)) + re(asin(pi)))*(I*im(asin(pi - asin(pi))) + re(asin(pi - asin(pi))))*(I*im(asin(asin(pi))) + re(asin(asin(pi))))

$$0 \pi \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\pi \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\pi \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\pi - \operatorname{asin}{\left(\pi \right)} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\pi - \operatorname{asin}{\left(\pi \right)} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\operatorname{asin}{\left(\pi \right)} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\operatorname{asin}{\left(\pi \right)} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\pi \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\pi \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\pi - \operatorname{asin}{\left(\pi \right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\pi - \operatorname{asin}{\left(\pi \right)} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\operatorname{asin}{\left(\pi \right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\operatorname{asin}{\left(\pi \right)} \right)}\right)}\right)$$

=

0

$$0$$

Численный ответ

x1 = 15.707963267949

x2 = 50.2654824574367

x3 = -56.5486677646163

x4 = 9.42477796076938

x5 = -53.4070751110265

x6 = 6.28318530717959

x7 = 84.8230016469244

x8 = -43.9822971502571

x9 = -62.8318530717959

x10 = 72.2566310325652

x11 = -50.2654824574367

x12 = -65.9734457253857

x13 = -12.5663706143592

x14 = -94.2477796076938

x15 = 31.4159265358979

x16 = -40.8407044966673

x17 = 56.5486677646163

x18 = 18.8495559215388

x19 = -34.5575191894877

x20 = 91.106186954104

x21 = 25.1327412287183

x22 = -81.6814089933346

x23 = 0.0

x24 = -87.9645943005142

x25 = -78.5398163397448

x26 = -31.4159265358979

x27 = 65.9734457253857

x28 = 47.1238898038469

x29 = -69.1150383789755

x30 = 1793.84940519977

x31 = 81.6814089933346

x32 = -59.6902604182061

x33 = 87.9645943005142

x34 = -6.28318530717959

x35 = -91.106186954104

x36 = -3.14159265358979

x37 = 37.6991118430775

x38 = -15.707963267949

x39 = -37.6991118430775

x40 = -100.530964914873

x41 = -75.398223686155

x42 = -9.42477796076938

x43 = 78.5398163397448

x44 = 69.1150383789755

x45 = 75.398223686155

x46 = 34.5575191894877

x47 = 12.5663706143592

x48 = -97.3893722612836

x49 = -28.2743338823081

x50 = 28.2743338823081

x51 = 62.8318530717959

x52 = -72.2566310325652

x53 = -125.663706143592

x54 = 43.9822971502571

x55 = -1668.18569905618

x56 = 21.9911485751286

x57 = -18.8495559215388

x58 = 59.6902604182061

x59 = -21.9911485751286

x60 = 97.3893722612836

x61 = -25.1327412287183

x62 = 53.4070751110265

x63 = -474.380490692059

x64 = -47.1238898038469

x65 = 100.530964914873

x66 = 3204.42450666159

x67 = 3.14159265358979

x68 = 40.8407044966673

x69 = 94.2477796076938

x70 = -84.8230016469244

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