cos2x=-1/2 (уравнение)

Найду корень уравнения: cos2x=-1/2

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:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{3}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{6}$$

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  

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

Быстрый ответ

     pi
x1 = --
     3 

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

     2*pi
x2 = ----
      3  

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

Численный ответ

x1 = -378.038315981972

x2 = 74.3510261349584

x3 = 82.7286065445312

x4 = -55.5014702134197

x5 = 38.7463093942741

x6 = 83.7758040957278

x7 = 33.5103216382911

x8 = -99.4837673636768

x9 = -96.342174710087

x10 = 26.1799387799149

x11 = -33.5103216382911

x12 = -5.23598775598299

x13 = -85.870199198121

x14 = -76.4454212373516

x15 = -32.4631240870945

x16 = 48.1710873550435

x17 = -90.0589894029074

x18 = 39.7935069454707

x19 = 68.0678408277789

x20 = -30.3687289847013

x21 = 99.4837673636768

x22 = -71.2094334813686

x23 = -294.262511886244

x24 = 10.471975511966

x25 = 2.0943951023932

x26 = 77.4926187885482

x27 = -79.5870138909414

x28 = -26.1799387799149

x29 = 54.4542726622231

x30 = -48.1710873550435

x31 = -49.2182849062401

x32 = 55.5014702134197

x33 = 98.4365698124802

x34 = -92.1533845053006

x35 = -4.18879020478639

x36 = -98.4365698124802

x37 = -39.7935069454707

x38 = -27.2271363311115

x39 = -10.471975511966

x40 = -63.8790506229925

x41 = -24.0855436775217

x42 = 16.7551608191456

x43 = 90.0589894029074

x44 = 1515.29485658148

x45 = -35.6047167406843

x46 = 96.342174710087

x47 = 63.8790506229925

x48 = 8.37758040957278

x49 = 24.0855436775217

x50 = -13.6135681655558

x51 = 19.8967534727354

x52 = 61.7846555205993

x53 = 85.870199198121

x54 = -61.7846555205993

x55 = 17.8023583703422

x56 = -19.8967534727354

x57 = -41.8879020478639

x58 = 60.7374579694027

x59 = 41.8879020478639

x60 = -54.4542726622231

x61 = -46.0766922526503

x62 = 52.3598775598299

x63 = -77.4926187885482

x64 = 70.162235930172

x65 = -2.0943951023932

x66 = 4.18879020478639

x67 = 76.4454212373516

x68 = -70.162235930172

x69 = -57.5958653158129

x70 = -11.5191730631626

x71 = 32.4631240870945

x72 = -17.8023583703422

x73 = 20.943951023932

x74 = -68.0678408277789

x75 = -14.6607657167524

x76 = 30.3687289847013

x77 = 11.5191730631626

x78 = -93.2005820564972

x79 = 92.1533845053006

x80 = 46.0766922526503

x81 = -83.7758040957278

x82 = 79.5870138909414

x83 = -60.7374579694027

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