Сумма корней cos(x)^3-cos(x)^2+cos(x)=1/3
Решение
Сумма и произведение корней
[src]
сумма
/ / / ___\\\ / / / ___\\\ / / / ___\\\ / / / ___\\\ / / / ___\\\ / / / ___\\\ / / / ___\\\ / / / ___\\\
| | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||
| | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - - -------||| | | 3 ___ 2 *|- - - -------||| / 3 ___ 2/3\ | | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - - -------||| | | 3 ___ 2 *|- - - -------||| / 3 ___ 2/3\
| |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /|| |1 \/ 2 2 | | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /|| |1 \/ 2 2 |
- re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|| + - re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|| + - acos|- - ----- + ----| + 2*pi + I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|| + I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|| + acos|- - ----- + ----|
| |3 / ___\ 3 || | |3 / ___\ 3 || | |3 / ___\ 3 || | |3 / ___\ 3 || \3 3 3 / | |3 / ___\ 3 || | |3 / ___\ 3 || | |3 / ___\ 3 || | |3 / ___\ 3 || \3 3 3 /
| | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||
| | 3*|- - + -------| || | | 3*|- - + -------| || | | 3*|- - - -------| || | | 3*|- - - -------| || | | 3*|- - + -------| || | | 3*|- - + -------| || | | 3*|- - - -------| || | | 3*|- - - -------| ||
\ \ \ 2 2 / // \ \ \ 2 2 / // \ \ \ 2 2 / // \ \ \ 2 2 / // \ \ \ 2 2 / // \ \ \ 2 2 / // \ \ \ 2 2 / // \ \ \ 2 2 / //
$$\operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + \left(\left(\left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) + \left(\left(- \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + 2 \pi\right) + \left(\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right)\right)\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right)\right)$$
=
6*pi
$$6 \pi$$
произведение
/ / / / ___\\\ / / / ___\\\\ / / / / ___\\\ / / / ___\\\\ / / / / ___\\\ / / / ___\\\\ / / / / ___\\\ / / / ___\\\\ | | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||| | | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||| | | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||| | | | 2/3 | 1 I*\/ 3 ||| | | 2/3 | 1 I*\/ 3 |||| | | | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - + -------|||| | | | 3 ___ 2 *|- - - -------||| | | 3 ___ 2 *|- - - -------|||| / / 3 ___ 2/3\ \ | | | 3 ___ 2 *|- - + -------||| | | 3 ___ 2 *|- - + -------|||| | | | 3 ___ 2 *|- - - -------||| | | 3 ___ 2 *|- - - -------|||| / 3 ___ 2/3\ | | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /||| | | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /||| | |1 \/ 2 2 | | | | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /||| | | |1 \/ 2 \ 2 2 /|| | |1 \/ 2 \ 2 2 /||| |1 \/ 2 2 | |- re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|||*|- re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|||*|- acos|- - ----- + ----| + 2*pi|*|I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|||*|I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|||*acos|- - ----- + ----| | | |3 / ___\ 3 || | |3 / ___\ 3 ||| | | |3 / ___\ 3 || | |3 / ___\ 3 ||| \ \3 3 3 / / | | |3 / ___\ 3 || | |3 / ___\ 3 ||| | | |3 / ___\ 3 || | |3 / ___\ 3 ||| \3 3 3 / | | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||| | | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||| | | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||| | | | | 1 I*\/ 3 | || | | | 1 I*\/ 3 | ||| | | | 3*|- - + -------| || | | 3*|- - + -------| ||| | | | 3*|- - - -------| || | | 3*|- - - -------| ||| | | | 3*|- - + -------| || | | 3*|- - + -------| ||| | | | 3*|- - - -------| || | | 3*|- - - -------| ||| \ \ \ \ 2 2 / // \ \ \ 2 2 / /// \ \ \ \ 2 2 / // \ \ \ 2 2 / /// \ \ \ \ 2 2 / // \ \ \ 2 2 / /// \ \ \ \ 2 2 / // \ \ \ 2 2 / ///
$$\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right) \left(- \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + 2 \pi\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right) \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}$$
=
/ / 3 ___ 2/3\ \ / / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\ / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\\ / / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\ / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\\ / / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\ / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\\ / / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\ / / 2/3 3 ___ 3 ___ ___ 2/3 ___\\\ / 3 ___ 2/3\ | |1 \/ 2 2 | | | | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 || | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 ||| | | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 || | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 ||| | | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 || | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 ||| | | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 || | |1 2 \/ 2 I*\/ 2 *\/ 3 I*2 *\/ 3 ||| |1 \/ 2 2 | |- acos|- - ----- + ----| + 2*pi|*|I*im|acos|- - ---- + ----- - ------------- - ------------|| + re|acos|- - ---- + ----- - ------------- - ------------|||*|I*im|acos|- - ---- + ----- + ------------- + ------------|| + re|acos|- - ---- + ----- + ------------- + ------------|||*|-2*pi + I*im|acos|- - ---- + ----- - ------------- - ------------|| + re|acos|- - ---- + ----- - ------------- - ------------|||*|-2*pi + I*im|acos|- - ---- + ----- + ------------- + ------------|| + re|acos|- - ---- + ----- + ------------- + ------------|||*acos|- - ----- + ----| \ \3 3 3 / / \ \ \3 6 6 6 6 // \ \3 6 6 6 6 /// \ \ \3 6 6 6 6 // \ \3 6 6 6 6 /// \ \ \3 6 6 6 6 // \ \3 6 6 6 6 /// \ \ \3 6 6 6 6 // \ \3 6 6 6 6 /// \3 3 3 /
$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)}\right) \left(- \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + 2 \pi\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)}\right) \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}$$