sin(a+t)=sin(a) (уравнение)

Найду корень уравнения: sin(a+t)=sin(a)

Вы ввели

sin(a + t) = sin(a)

$$\sin{\left(a + t \right)} = \sin{\left(a \right)}$$

Подробное решение

Дано уравнение
$$\sin{\left(a + t \right)} = \sin{\left(a \right)}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$a + t = 2 \pi n + \operatorname{asin}{\left(\sin{\left(a \right)} \right)}$$
$$a + t = 2 \pi n - \operatorname{asin}{\left(\sin{\left(a \right)} \right)} + \pi$$
Или
$$a + t = 2 \pi n + \operatorname{asin}{\left(\sin{\left(a \right)} \right)}$$
$$a + t = 2 \pi n - \operatorname{asin}{\left(\sin{\left(a \right)} \right)} + \pi$$
, где n - любое целое число
Перенесём
$$a$$
в правую часть ур-ния
с противоположным знаком, итого:
$$t = - a + 2 \pi n + \operatorname{asin}{\left(\sin{\left(a \right)} \right)}$$
$$t = - a + 2 \pi n - \operatorname{asin}{\left(\sin{\left(a \right)} \right)} + \pi$$

Быстрый ответ

t1 = -re(a) + I*(-im(a) + im(asin(sin(a)))) + re(asin(sin(a)))

$$t_{1} = i \left(- \operatorname{im}{\left(a\right)} + \operatorname{im}{\left(\operatorname{asin}{\left(\sin{\left(a \right)} \right)}\right)}\right) - \operatorname{re}{\left(a\right)} + \operatorname{re}{\left(\operatorname{asin}{\left(\sin{\left(a \right)} \right)}\right)}$$

t2 = pi - re(a) - re(asin(sin(a))) + I*(-im(a) - im(asin(sin(a))))

$$t_{2} = i \left(- \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(\operatorname{asin}{\left(\sin{\left(a \right)} \right)}\right)}\right) - \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(\operatorname{asin}{\left(\sin{\left(a \right)} \right)}\right)} + \pi$$

Сумма и произведение корней

сумма

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

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

=

pi - 2*re(a) + I*(-im(a) - im(asin(sin(a)))) + I*(-im(a) + im(asin(sin(a))))

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

произведение

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

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

=

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

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