cos(4pi*x/3)=-1/2 (уравнение)

Найду корень уравнения: cos(4pi*x/3)=-1/2

Вы ввели

   /4*pi*x\       
cos|------| = -1/2
   \  3   /       

$$\cos{\left(\frac{4 \pi x}{3} \right)} = - \frac{1}{2}$$

Подробное решение

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

Быстрый ответ

x1 = 1/2

$$x_{1} = \frac{1}{2}$$

x2 = 1

$$x_{2} = 1$$

Сумма и произведение корней

сумма

0 + 1/2 + 1

$$\left(0 + \frac{1}{2}\right) + 1$$

=

3/2

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

произведение

1*1/2*1

$$1 \cdot \frac{1}{2} \cdot 1$$

=

1/2

$$\frac{1}{2}$$

Численный ответ

x1 = 82.0

x2 = 46.0

x3 = -17.5

x4 = 6.5

x5 = -76.0

x6 = 2.0

x7 = 50.0

x8 = 70.0

x9 = -2.0

x10 = -77.5

x11 = 100.0

x12 = -34.0

x13 = -52.0

x14 = 18.5

x15 = -40.0

x16 = 8.0

x17 = 88.0

x18 = -5.5

x19 = 0.5

x20 = -70.0

x21 = 56.0

x22 = -46.0

x23 = -56.0

x24 = -11.5

x25 = -44.0

x26 = -23.5

x27 = 52.0

x28 = 38.0

x29 = -28.0

x30 = -47.5

x31 = -80.0

x32 = 44.0

x33 = 40.0

x34 = -38.0

x35 = 28.0

x36 = -74.0

x37 = 94.0

x38 = 48.5

x39 = -95.5

x40 = -35.5

x41 = 66.5

x42 = -89.5

x43 = 90.5

x44 = 58.0

x45 = -20.0

x46 = -88.0

x47 = -62.0

x48 = -29.5

x49 = 30.5

x50 = -71.5

x51 = 76.0

x52 = 22.0

x53 = 54.5

x54 = 92.0

x55 = -14.0

x56 = -92.0

x57 = 80.0

x58 = 14.0

x59 = 60.5

x60 = 36.5

x61 = 62.0

x62 = 98.0

x63 = -26.0

x64 = 10.0

x65 = -82.0

x66 = -10.0

x67 = -53.5

x68 = 78.5

x69 = -8.0

x70 = -86.0

x71 = -41.5

x72 = 64.0

x73 = -32.0

x74 = -64.0

x75 = 32.0

x76 = 74.0

x77 = 72.5

x78 = -58.0

x79 = -16.0

x80 = -68.0

x81 = -83.5

x82 = 68.0

x83 = 86.0

x84 = 84.5

x85 = -50.0

x86 = 26.0

x87 = -65.5

x88 = -59.5

x89 = 20.0

x90 = -94.0

x91 = 34.0

x92 = -22.0

x93 = 96.5

x94 = 42.5

x95 = 16.0

x96 = -4.0

x97 = -100.0

x98 = 24.5

x99 = 12.5

x100 = 4.0

x101 = -98.0

График