Произведение корней cos(4*pi*x)/3=-1/2
Решение
Сумма и произведение корней
[src]
сумма
-re(acos(-3/2)) + 2*pi I*im(acos(-3/2)) re(acos(-3/2)) I*im(acos(-3/2))
---------------------- - ---------------- + -------------- + ----------------
4*pi 4*pi 4*pi 4*pi
$$\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}}{4 \pi} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}}{4 \pi}\right) + \left(\frac{- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 2 \pi}{4 \pi} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}}{4 \pi}\right)$$
=
-re(acos(-3/2)) + 2*pi re(acos(-3/2))
---------------------- + --------------
4*pi 4*pi
$$\frac{- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 2 \pi}{4 \pi} + \frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}}{4 \pi}$$
произведение
/-re(acos(-3/2)) + 2*pi I*im(acos(-3/2))\ /re(acos(-3/2)) I*im(acos(-3/2))\ |---------------------- - ----------------|*|-------------- + ----------------| \ 4*pi 4*pi / \ 4*pi 4*pi /
$$\left(\frac{- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 2 \pi}{4 \pi} - \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}}{4 \pi}\right) \left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}}{4 \pi} + \frac{i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}}{4 \pi}\right)$$
=
(I*im(acos(-3/2)) + re(acos(-3/2)))*(-re(acos(-3/2)) + 2*pi - I*im(acos(-3/2)))
-------------------------------------------------------------------------------
2
16*pi
$$\frac{\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{3}{2} \right)}\right)}\right)}{16 \pi^{2}}$$