cos(pi*(x-7)/3)=0.5 (уравнение)

Найду корень уравнения: cos(pi*(x-7)/3)=0.5

Вы ввели

   /pi*(x - 7)\      
cos|----------| = 1/2
   \    3     /      

$$\cos{\left(\frac{\pi \left(x - 7\right)}{3} \right)} = \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(\frac{\pi \left(x - 7\right)}{3} \right)} = \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$\frac{\pi x}{3} + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$\frac{\pi x}{3} + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Или
$$\frac{\pi x}{3} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{\pi x}{3} + \frac{\pi}{6} = 2 \pi n + \frac{5 \pi}{6}$$
, где n - любое целое число
Перенесём
$$\frac{\pi}{6}$$
в правую часть ур-ния
с противоположным знаком, итого:
$$\frac{\pi x}{3} = 2 \pi n$$
$$\frac{\pi x}{3} = 2 \pi n + \frac{2 \pi}{3}$$
Разделим обе части полученного ур-ния на
$$\frac{\pi}{3}$$
получим ответ:
$$x_{1} = 6 n$$
$$x_{2} = \frac{3 \left(2 \pi n + \frac{2 \pi}{3}\right)}{\pi}$$

Быстрый ответ

x1 = 0

$$x_{1} = 0$$

x2 = 2

$$x_{2} = 2$$

Сумма и произведение корней

сумма

2

$$2$$

=

2

$$2$$

произведение

0*2

$$0 \cdot 2$$

=

0

$$0$$

Численный ответ

x1 = 48.0

x2 = -24.0

x3 = 30.0

x4 = -90.0

x5 = 644.0

x6 = -64.0

x7 = -876.0

x8 = 14.0

x9 = -78.0

x10 = 68.0

x11 = 54.0

x12 = 62.0

x13 = -88.0

x14 = 66.0

x15 = -46.0

x16 = 24.0

x17 = -6.0

x18 = 44.0

x19 = -30.0

x20 = 32.0

x21 = -4.0

x22 = -96.0

x23 = -22.0

x24 = 20.0

x25 = -40.0

x26 = -48.0

x27 = 86.0

x28 = 42.0

x29 = -42.0

x30 = -12.0

x31 = -82.0

x32 = -28.0

x33 = 600.0

x34 = -72.0

x35 = 12.0

x36 = 8.0

x37 = 78.0

x38 = -36.0

x39 = -16.0

x40 = 98.0

x41 = -58.0

x42 = 74.0

x43 = 56.0

x44 = 26.0

x45 = -70.0

x46 = 18.0

x47 = 92.0

x48 = 488.0

x49 = -60.0

x50 = -54.0

x51 = 90.0

x52 = -66.0

x53 = 446.0

x54 = 80.0

x55 = 6.0

x56 = 2.0

x57 = 60.0

x58 = -34.0

x59 = 752.0

x60 = 72.0

x61 = 38.0

x62 = -10.0

x63 = -94.0

x64 = -93744.0

x65 = 50.0

x66 = -76.0

x67 = 36.0

x68 = -52.0

x69 = -84.0

x70 = -100.0

x71 = -18.0

x72 = 84.0

x73 = 96.0

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