cos(4*pi*x/3)=sqrt(3)/2 (уравнение)

Найду корень уравнения: cos(4*pi*x/3)=sqrt(3)/2

Вы ввели

                ___
   /4*pi*x\   \/ 3 
cos|------| = -----
   \  3   /     2  

$$\cos{\left(\frac{4 \pi x}{3} \right)} = \frac{\sqrt{3}}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(\frac{4 \pi x}{3} \right)} = \frac{\sqrt{3}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$\frac{4 \pi x}{3} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$\frac{4 \pi x}{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}$$
Или
$$\frac{4 \pi x}{3} = \pi n + \frac{\pi}{6}$$
$$\frac{4 \pi x}{3} = \pi n - \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$\frac{4 \pi}{3}$$
получим ответ:
$$x_{1} = \frac{3 \left(\pi n + \frac{\pi}{6}\right)}{4 \pi}$$
$$x_{2} = \frac{3 \left(\pi n - \frac{5 \pi}{6}\right)}{4 \pi}$$

Быстрый ответ

x1 = 1/8

$$x_{1} = \frac{1}{8}$$

x2 = 11/8

$$x_{2} = \frac{11}{8}$$

Сумма и произведение корней

сумма

0 + 1/8 + 11/8

$$\left(0 + \frac{1}{8}\right) + \frac{11}{8}$$

=

3/2

$$\frac{3}{2}$$

произведение

1*1/8*11/8

$$1 \cdot \frac{1}{8} \cdot \frac{11}{8}$$

=

11
--
64

$$\frac{11}{64}$$

Численный ответ

x1 = -5.875

x2 = -49.625

x3 = 58.375

x4 = 90.125

x5 = 7.375

x6 = -29.875

x7 = 10.375

x8 = -1.625

x9 = 96.125

x10 = -23.875

x11 = 97.375

x12 = -55.625

x13 = -11.875

x14 = -76.625

x15 = 34.375

x16 = -31.625

x17 = -91.625

x18 = 66.125

x19 = 30.125

x20 = -10.625

x21 = 36.125

x22 = -67.625

x23 = -47.875

x24 = 48.125

x25 = -61.625

x26 = -89.875

x27 = -58.625

x28 = -35.875

x29 = -94.625

x30 = 52.375

x31 = 64.375

x32 = -46.625

x33 = 82.375

x34 = -71.875

x35 = -34.625

x36 = 12.125

x37 = -13.625

x38 = -79.625

x39 = 79.375

x40 = -95.875

x41 = 84.125

x42 = -43.625

x43 = 100.375

x44 = 13.375

x45 = -77.875

x46 = -100.625

x47 = -70.625

x48 = -59.875

x49 = 94.375

x50 = -52.625

x51 = -82.625

x52 = 22.375

x53 = 76.375

x54 = -25.625

x55 = 43.375

x56 = -65.875

x57 = -37.625

x58 = -40.625

x59 = 73.375

x60 = 70.375

x61 = -22.625

x62 = -19.625

x63 = -41.875

x64 = 25.375

x65 = 42.125

x66 = 28.375

x67 = -53.875

x68 = -97.625

x69 = 61.375

x70 = 6.125

x71 = -28.625

x72 = -7.625

x73 = 46.375

x74 = 85.375

x75 = -85.625

x76 = -83.875

x77 = -88.625

x78 = 19.375

x79 = -16.625

x80 = 24.125

x81 = 72.125

x82 = -64.625

x83 = 88.375

x84 = 16.375

x85 = 18.125

x86 = -73.625

x87 = 40.375

x88 = -17.875

x89 = 67.375

x90 = 54.125

x91 = 49.375

x92 = 60.125

x93 = 37.375

x94 = 78.125

x95 = 1.375

x96 = 4.375

x97 = -4.625

x98 = 31.375

x99 = 91.375

x100 = 55.375

x101 = 0.125

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