cos2x=√2/2 (уравнение)

Найду корень уравнения: cos2x=√2/2

Вы ввели

             ___
           \/ 2 
cos(2*x) = -----
             2  

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

Подробное решение

Дано уравнение
$$\cos{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$2 x = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
Или
$$2 x = \pi n + \frac{\pi}{4}$$
$$2 x = \pi n - \frac{3 \pi}{4}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$2$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
$$x_{2} = \frac{\pi n}{2} - \frac{3 \pi}{8}$$

Быстрый ответ

     pi
x1 = --
     8 

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

     7*pi
x2 = ----
      8  

$$x_{2} = \frac{7 \pi}{8}$$

Сумма и произведение корней

сумма

    pi   7*pi
0 + -- + ----
    8     8  

$$\left(0 + \frac{\pi}{8}\right) + \frac{7 \pi}{8}$$

=

pi

$$\pi$$

произведение

  pi 7*pi
1*--*----
  8   8  

$$\frac{7 \pi}{8} \cdot 1 \frac{\pi}{8}$$

=

    2
7*pi 
-----
  64 

$$\frac{7 \pi^{2}}{64}$$

Численный ответ

x1 = 62.4391539900971

x2 = -87.5718952188155

x3 = 109.563043793944

x4 = 97.7820713429823

x5 = 91.4988860358027

x6 = -62.4391539900971

x7 = -82.0741080750334

x8 = -85.2157007286231

x9 = 60.0829594999048

x10 = -2.74889357189107

x11 = 22.3838476568273

x12 = -12.1736715326604

x13 = 49.872783375738

x14 = -47.5165888855456

x15 = 46.7311907221482

x16 = 82.0741080750334

x17 = -43.5895980685584

x18 = 68.7223392972767

x19 = -56.1559686829176

x20 = -46.7311907221482

x21 = 44.3749962319558

x22 = -103.279858486764

x23 = -18.45685683984

x24 = 69.5077374606742

x25 = -16.1006623496477

x26 = 90.7134878724053

x27 = 93.8550805259951

x28 = -41.233403578366

x29 = -60.0829594999048

x30 = 40.4480054149686

x31 = -3.53429173528852

x32 = 18.45685683984

x33 = 2.74889357189107

x34 = -9.8174770424681

x35 = 27.8816348006094

x36 = -65.5807466436869

x37 = 56.1559686829176

x38 = 34.164820107789

x39 = 78.1471172580461

x40 = 47.5165888855456

x41 = 0.392699081698724

x42 = -34.164820107789

x43 = 24.7400421470196

x44 = -75.7909227678538

x45 = -97.7820713429823

x46 = -90.7134878724053

x47 = -24.7400421470196

x48 = -426.863901806513

x49 = 38.0918109247762

x50 = 3.53429173528852

x51 = -19.2422550032375

x52 = 5.89048622548086

x53 = -71.8639319508665

x54 = -27.8816348006094

x55 = -38.0918109247762

x56 = 75.7909227678538

x57 = -286.27763055837

x58 = -78.1471172580461

x59 = -21.5984494934298

x60 = -31.8086256175967

x61 = -40.4480054149686

x62 = -5.89048622548086

x63 = -63.2245521534946

x64 = -69.5077374606742

x65 = 66.3661448070844

x66 = 25.5254403104171

x67 = 9.8174770424681

x68 = -68.7223392972767

x69 = -84.4303025652257

x70 = 71.8639319508665

x71 = 53.7997741927252

x72 = 100.138265833175

x73 = 88.3572933822129

x74 = -25.5254403104171

x75 = -53.7997741927252

x76 = 96.9966731795849

x77 = 12.1736715326604

x78 = -93.8550805259951

x79 = 138.62277583965

x80 = -91.4988860358027

x81 = 84.4303025652257

x82 = -49.872783375738

x83 = -109.563043793944

x84 = 16.1006623496477

x85 = -100.138265833175

x86 = 31.8086256175967

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