cos(2x)=-0,5 (уравнение)

Найду корень уравнения: cos(2x)=-0,5

Вы ввели

cos(2*x) = -1/2

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

Подробное решение

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

Быстрый ответ

     pi
x1 = --
     3 

$$x_{1} = \frac{\pi}{3}$$

     2*pi
x2 = ----
      3  

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

Сумма и произведение корней

сумма

    pi   2*pi
0 + -- + ----
    3     3  

$$\left(0 + \frac{\pi}{3}\right) + \frac{2 \pi}{3}$$

=

pi

$$\pi$$

произведение

  pi 2*pi
1*--*----
  3   3  

$$\frac{2 \pi}{3} \cdot 1 \frac{\pi}{3}$$

=

    2
2*pi 
-----
  9  

$$\frac{2 \pi^{2}}{9}$$

Численный ответ

x1 = 77.4926187885482

x2 = -10.471975511966

x3 = -92.1533845053006

x4 = -99.4837673636768

x5 = -378.038315981972

x6 = 16.7551608191456

x7 = 48.1710873550435

x8 = -14.6607657167524

x9 = -294.262511886244

x10 = -32.4631240870945

x11 = 90.0589894029074

x12 = 26.1799387799149

x13 = 41.8879020478639

x14 = 38.7463093942741

x15 = -96.342174710087

x16 = -83.7758040957278

x17 = 92.1533845053006

x18 = -77.4926187885482

x19 = -46.0766922526503

x20 = 85.870199198121

x21 = 83.7758040957278

x22 = 33.5103216382911

x23 = 1515.29485658148

x24 = -4.18879020478639

x25 = 8.37758040957278

x26 = -17.8023583703422

x27 = 30.3687289847013

x28 = 10.471975511966

x29 = 82.7286065445312

x30 = 98.4365698124802

x31 = -93.2005820564972

x32 = -61.7846555205993

x33 = 68.0678408277789

x34 = 79.5870138909414

x35 = -57.5958653158129

x36 = -79.5870138909414

x37 = 55.5014702134197

x38 = -60.7374579694027

x39 = -76.4454212373516

x40 = -19.8967534727354

x41 = 4.18879020478639

x42 = -85.870199198121

x43 = 74.3510261349584

x44 = 32.4631240870945

x45 = -70.162235930172

x46 = 17.8023583703422

x47 = 52.3598775598299

x48 = -54.4542726622231

x49 = 46.0766922526503

x50 = 60.7374579694027

x51 = -33.5103216382911

x52 = 61.7846555205993

x53 = -71.2094334813686

x54 = -55.5014702134197

x55 = 11.5191730631626

x56 = -39.7935069454707

x57 = 54.4542726622231

x58 = 99.4837673636768

x59 = -98.4365698124802

x60 = -26.1799387799149

x61 = -13.6135681655558

x62 = 63.8790506229925

x63 = 96.342174710087

x64 = 19.8967534727354

x65 = -90.0589894029074

x66 = -30.3687289847013

x67 = -49.2182849062401

x68 = 76.4454212373516

x69 = -63.8790506229925

x70 = -48.1710873550435

x71 = 39.7935069454707

x72 = 24.0855436775217

x73 = 70.162235930172

x74 = -35.6047167406843

x75 = -68.0678408277789

x76 = -24.0855436775217

x77 = -11.5191730631626

x78 = 20.943951023932

x79 = -27.2271363311115

x80 = -5.23598775598299

x81 = -41.8879020478639

x82 = -2.0943951023932

x83 = 2.0943951023932

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