cos(pi*x-pi/4)=0 (уравнение)

Найду корень уравнения: cos(pi*x-pi/4)=0

Вы ввели

   /       pi\    
cos|pi*x - --| = 0
   \       4 /    

$$\cos{\left(\pi x - \frac{\pi}{4} \right)} = 0$$

Подробное решение

Дано уравнение
$$\cos{\left(\pi x - \frac{\pi}{4} \right)} = 0$$
- это простейшее тригонометрическое ур-ние

с изменением знака при 0

Получим:
$$\cos{\left(\pi x - \frac{\pi}{4} \right)} = 0$$
Это ур-ние преобразуется в
$$\pi x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\pi x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Или
$$\pi x + \frac{\pi}{4} = 2 \pi n$$
$$\pi x + \frac{\pi}{4} = 2 \pi n + \pi$$
, где n - любое целое число
Перенесём
$$\frac{\pi}{4}$$
в правую часть ур-ния
с противоположным знаком, итого:
$$\pi x = 2 \pi n - \frac{\pi}{4}$$
$$\pi x = 2 \pi n + \frac{3 \pi}{4}$$
Разделим обе части полученного ур-ния на
$$\pi$$
получим ответ:
$$x_{1} = \frac{2 \pi n - \frac{\pi}{4}}{\pi}$$
$$x_{2} = \frac{2 \pi n + \frac{3 \pi}{4}}{\pi}$$

Быстрый ответ

x1 = -1/4

$$x_{1} = - \frac{1}{4}$$

x2 = 3/4

$$x_{2} = \frac{3}{4}$$

Сумма и произведение корней

сумма

0 - 1/4 + 3/4

$$\left(- \frac{1}{4} + 0\right) + \frac{3}{4}$$

=

1/2

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

произведение

1*-1/4*3/4

$$1 \left(- \frac{1}{4}\right) \frac{3}{4}$$

=

-3/16

$$- \frac{3}{16}$$

Численный ответ

x1 = -90.25

x2 = -78.25

x3 = 7.75

x4 = -10.25

x5 = -8.25

x6 = -46.25

x7 = -44.25

x8 = -54.25

x9 = 19.75

x10 = 73.75

x11 = 45.75

x12 = -84.25

x13 = -4.25

x14 = -74.25

x15 = -2.25

x16 = 99.75

x17 = 81.75

x18 = -98.25

x19 = -24.25

x20 = 89.75

x21 = 23.75

x22 = -60.25

x23 = -22.25

x24 = -88.25

x25 = -6.25

x26 = 61.75

x27 = 25.75

x28 = 13.75

x29 = -58.25

x30 = -38.25

x31 = 51.75

x32 = 91.75

x33 = 85.75

x34 = 39.75

x35 = -42.25

x36 = -68.25

x37 = -50.25

x38 = 87.75

x39 = -96.25

x40 = 93.75

x41 = 67.75

x42 = -26.25

x43 = 1.75

x44 = -14.25

x45 = -20.25

x46 = -82.25

x47 = 71.75

x48 = -94.25

x49 = -28.25

x50 = -76.25

x51 = 5.75

x52 = -72.25

x53 = 3.75

x54 = -32.25

x55 = 79.75

x56 = -70.25

x57 = 49.75

x58 = -0.25

x59 = 9.75

x60 = 11.75

x61 = 69.75

x62 = -36.25

x63 = 41.75

x64 = 95.75

x65 = -30.25

x66 = -34.25

x67 = 21.75

x68 = 53.75

x69 = -62.25

x70 = 43.75

x71 = 31.75

x72 = 33.75

x73 = -16.25

x74 = 37.75

x75 = -100.25

x76 = 35.75

x77 = -18.25

x78 = -92.25

x79 = -40.25

x80 = 55.75

x81 = 57.75

x82 = 97.75

x83 = -66.25

x84 = -52.25

x85 = 27.75

x86 = 47.75

x87 = 83.75

x88 = -86.25

x89 = 15.75

x90 = 29.75

x91 = -64.25

x92 = 63.75

x93 = -80.25

x94 = 17.75

x95 = -48.25

x96 = 77.75

x97 = -56.25

x98 = 75.75

x99 = 59.75

x100 = 65.75

x101 = -12.25

График