cos(pi*x) = -i (уравнение)

Найду корень уравнения: cos(pi*x) = -i

Вы ввели

cos(pi*x) = -I

$$\cos{\left(\pi x \right)} = - i$$

Подробное решение

Дано уравнение
$$\cos{\left(\pi x \right)} = - i$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$\pi x = \pi n + \operatorname{acos}{\left(- i \right)}$$
$$\pi x = \pi n - \pi + \operatorname{acos}{\left(- i \right)}$$
Или
$$\pi x = \pi n + \frac{\pi}{2} + i \log{\left(1 + \sqrt{2} \right)}$$
$$\pi x = \pi n - \frac{\pi}{2} + i \log{\left(1 + \sqrt{2} \right)}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$\pi$$
получим ответ:
$$x_{1} = \frac{\pi n + \frac{\pi}{2} + i \log{\left(1 + \sqrt{2} \right)}}{\pi}$$
$$x_{2} = \frac{\pi n - \frac{\pi}{2} + i \log{\left(1 + \sqrt{2} \right)}}{\pi}$$

Быстрый ответ

              /      ___\
     1   I*log\1 + \/ 2 /
x1 = - + ----------------
     2          pi       

$$x_{1} = \frac{1}{2} + \frac{i \log{\left(1 + \sqrt{2} \right)}}{\pi}$$

              /      ___\
     3   I*log\1 + \/ 2 /
x2 = - - ----------------
     2          pi       

$$x_{2} = \frac{3}{2} - \frac{i \log{\left(1 + \sqrt{2} \right)}}{\pi}$$

Численный ответ

x1 = 88.5 + 0.28054992616959*i

x2 = 10.5 + 0.28054992616959*i

x3 = -41.5 + 0.28054992616959*i

x4 = 64.5 + 0.28054992616959*i

x5 = -93.5 + 0.28054992616959*i

x6 = -31.5 + 0.28054992616959*i

x7 = 30.5 + 0.28054992616959*i

x8 = -35.5 + 0.28054992616959*i

x9 = 4.5 + 0.28054992616959*i

x10 = -33.5 + 0.28054992616959*i

x11 = 76.5 + 0.28054992616959*i

x12 = 98.5 + 0.28054992616959*i

x13 = 52.5 + 0.28054992616959*i

x14 = -3.5 + 0.28054992616959*i

x15 = -7.5 + 0.28054992616959*i

x16 = 18.5 + 0.28054992616959*i

x17 = 50.5 + 0.28054992616959*i

x18 = 6.5 + 0.28054992616959*i

x19 = -17.5 + 0.28054992616959*i

x20 = 26.5 + 0.28054992616959*i

x21 = -99.5 + 0.28054992616959*i

x22 = 78.5 + 0.28054992616959*i

x23 = 94.5 + 0.28054992616959*i

x24 = 82.5 + 0.28054992616959*i

x25 = 62.5 + 0.28054992616959*i

x26 = 8.5 + 0.28054992616959*i

x27 = 74.5 + 0.28054992616959*i

x28 = 54.5 + 0.28054992616959*i

x29 = -49.5 + 0.28054992616959*i

x30 = 12.5 + 0.28054992616959*i

x31 = -67.5 + 0.28054992616959*i

x32 = -39.5 + 0.28054992616959*i

x33 = 70.5 + 0.28054992616959*i

x34 = 32.5 + 0.28054992616959*i

x35 = -29.5 + 0.28054992616959*i

x36 = -13.5 + 0.28054992616959*i

x37 = -77.5 + 0.28054992616959*i

x38 = -85.5 + 0.28054992616959*i

x39 = 28.5 + 0.28054992616959*i

x40 = 100.5 + 0.28054992616959*i

x41 = 14.5 + 0.28054992616959*i

x42 = 84.5 + 0.28054992616959*i

x43 = 72.5 + 0.28054992616959*i

x44 = 96.5 + 0.28054992616959*i

x45 = -89.5 + 0.28054992616959*i

x46 = -47.5 + 0.28054992616959*i

x47 = -95.5 + 0.28054992616959*i

x48 = 92.5 + 0.28054992616959*i

x49 = 2.5 + 0.28054992616959*i

x50 = 20.5 + 0.28054992616959*i

x51 = -97.5 + 0.28054992616959*i

x52 = -79.5 + 0.28054992616959*i

x53 = -15.5 + 0.28054992616959*i

x54 = -19.5 + 0.28054992616959*i

x55 = -1.5 + 0.28054992616959*i

x56 = 16.5 + 0.28054992616959*i

x57 = -9.5 + 0.28054992616959*i

x58 = -91.5 + 0.28054992616959*i

x59 = 44.5 + 0.28054992616959*i

x60 = -73.5 + 0.28054992616959*i

x61 = -59.5 + 0.28054992616959*i

x62 = 46.5 + 0.28054992616959*i

x63 = -11.5 + 0.28054992616959*i

x64 = 42.5 + 0.28054992616959*i

x65 = 60.5 + 0.28054992616959*i

x66 = -51.5 + 0.28054992616959*i

x67 = 22.5 + 0.28054992616959*i

x68 = -5.5 + 0.28054992616959*i

x69 = 40.5 + 0.28054992616959*i

x70 = -57.5 + 0.28054992616959*i

x71 = -63.5 + 0.28054992616959*i

x72 = -87.5 + 0.28054992616959*i

x73 = 90.5 + 0.28054992616959*i

x74 = 68.5 + 0.28054992616959*i

x75 = -75.5 + 0.28054992616959*i

x76 = -21.5 + 0.28054992616959*i

x77 = -81.5 + 0.28054992616959*i

x78 = -69.5 + 0.28054992616959*i

x79 = -43.5 + 0.28054992616959*i

x80 = -23.5 + 0.28054992616959*i

x81 = 36.5 + 0.28054992616959*i

x82 = -25.5 + 0.28054992616959*i

x83 = -61.5 + 0.28054992616959*i

x84 = -55.5 + 0.28054992616959*i

x85 = 24.5 + 0.28054992616959*i

x86 = -65.5 + 0.28054992616959*i

x87 = 0.5 + 0.28054992616959*i

x88 = 86.5 + 0.28054992616959*i

x89 = -27.5 + 0.28054992616959*i

x90 = 80.5 + 0.28054992616959*i

x91 = -45.5 + 0.28054992616959*i

x92 = 38.5 + 0.28054992616959*i

x93 = 56.5 + 0.28054992616959*i

x94 = 58.5 + 0.28054992616959*i

x95 = -53.5 + 0.28054992616959*i

x96 = -37.5 + 0.28054992616959*i

x97 = -71.5 + 0.28054992616959*i

x98 = 48.5 + 0.28054992616959*i

x99 = 34.5 + 0.28054992616959*i

x100 = 66.5 + 0.28054992616959*i

x101 = -83.5 + 0.28054992616959*i