cos(x^2)=2 (уравнение)

Найду корень уравнения: cos(x^2)=2

Вы ввели

   / 2\    
cos\x / = 2

$$\cos{\left(x^{2} \right)} = 2$$

Подробное решение

Дано уравнение
$$\cos{\left(x^{2} \right)} = 2$$
преобразуем
$$\cos{\left(x^{2} \right)} - 2 = 0$$
$$\cos{\left(x^{2} \right)} - 2 = 0$$
Сделаем замену
$$w = \cos{\left(x^{2} \right)}$$
Переносим свободные слагаемые (без w)
из левой части в правую, получим:
$$w = 2$$
Получим ответ: w = 2
делаем обратную замену
$$\cos{\left(x^{2} \right)} = w$$
подставляем w:

Быстрый ответ

                                    /    /im(acos(2))\\                                  /    /im(acos(2))\\
          ______________________    |atan|-----------||        ______________________    |atan|-----------||
       4 /   2                2     |    \    2*pi   /|     4 /   2                2     |    \    2*pi   /|
x1 = - \/  im (acos(2)) + 4*pi  *cos|-----------------| + I*\/  im (acos(2)) + 4*pi  *sin|-----------------|
                                    \        2        /                                  \        2        /

$$x_{1} = - \sqrt[4]{\left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + 4 \pi^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{2 \pi} \right)}}{2} \right)} + i \sqrt[4]{\left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + 4 \pi^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{2 \pi} \right)}}{2} \right)}$$

                                  /    /im(acos(2))\\                                  /    /im(acos(2))\\
        ______________________    |atan|-----------||        ______________________    |atan|-----------||
     4 /   2                2     |    \    2*pi   /|     4 /   2                2     |    \    2*pi   /|
x2 = \/  im (acos(2)) + 4*pi  *cos|-----------------| - I*\/  im (acos(2)) + 4*pi  *sin|-----------------|
                                  \        2        /                                  \        2        /

$$x_{2} = \sqrt[4]{\left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + 4 \pi^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{2 \pi} \right)}}{2} \right)} - i \sqrt[4]{\left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + 4 \pi^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{2 \pi} \right)}}{2} \right)}$$

                                           /     /  I*im(acos(2)) + re(acos(2))   \\                                         /     /  I*im(acos(2)) + re(acos(2))   \\
                                           |I*log|--------------------------------||                                         |I*log|--------------------------------||
                                           |     |   _____________________________||                                         |     |   _____________________________||
          _____________________________    |     |  /   2              2          ||        _____________________________    |     |  /   2              2          ||
       4 /   2              2              |     \\/  im (acos(2)) + re (acos(2)) /|     4 /   2              2              |     \\/  im (acos(2)) + re (acos(2)) /|
x3 = - \/  im (acos(2)) + re (acos(2)) *cos|---------------------------------------| + I*\/  im (acos(2)) + re (acos(2)) *sin|---------------------------------------|
                                           \                   2                   /                                         \                   2                   /

$$x_{3} = - \sqrt[4]{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}} \cos{\left(\frac{i \log{\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{\sqrt{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}}} \right)}}{2} \right)} + i \sqrt[4]{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}} \sin{\left(\frac{i \log{\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{\sqrt{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}}} \right)}}{2} \right)}$$

                                         /     /  I*im(acos(2)) + re(acos(2))   \\                                         /     /  I*im(acos(2)) + re(acos(2))   \\
                                         |I*log|--------------------------------||                                         |I*log|--------------------------------||
                                         |     |   _____________________________||                                         |     |   _____________________________||
        _____________________________    |     |  /   2              2          ||        _____________________________    |     |  /   2              2          ||
     4 /   2              2              |     \\/  im (acos(2)) + re (acos(2)) /|     4 /   2              2              |     \\/  im (acos(2)) + re (acos(2)) /|
x4 = \/  im (acos(2)) + re (acos(2)) *cos|---------------------------------------| - I*\/  im (acos(2)) + re (acos(2)) *sin|---------------------------------------|
                                         \                   2                   /                                         \                   2                   /

$$x_{4} = \sqrt[4]{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}} \cos{\left(\frac{i \log{\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{\sqrt{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}}} \right)}}{2} \right)} - i \sqrt[4]{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}} \sin{\left(\frac{i \log{\left(\frac{\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}}{\sqrt{\left(\operatorname{re}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2} + \left(\operatorname{im}{\left(\operatorname{acos}{\left(2 \right)}\right)}\right)^{2}}} \right)}}{2} \right)}$$

Численный ответ

x1 = -2.52020878032318 + 0.261279523190126*i

x2 = 2.52020878032318 - 0.261279523190126*i

x3 = -0.811467157969075 - 0.811467157969075*i

x4 = 0.811467157969075 + 0.811467157969075*i

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