|cosx|=1/2 (уравнение)

Найду корень уравнения: |cosx|=1/2

Вы ввели

|cos(x)| = 1/2

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

Подробное решение

Дано уравнение
$$\left|{\cos{\left(x \right)}}\right| = \frac{1}{2}$$
преобразуем
$$\left|{\cos{\left(x \right)}}\right| - \frac{1}{2} = 0$$
$$\left|{\cos{\left(x \right)}}\right| - \frac{1}{2} = 0$$
Сделаем замену
$$w = \left|{\cos{\left(x \right)}}\right|$$
Переносим свободные слагаемые (без w)
из левой части в правую, получим:
$$w = \frac{1}{2}$$
Получим ответ: w = 1/2
делаем обратную замену
$$\left|{\cos{\left(x \right)}}\right| = w$$
подставляем w:

Быстрый ответ

     pi
x1 = --
     3 

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

     2*pi
x2 = ----
      3  

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

     4*pi
x3 = ----
      3  

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

     5*pi
x4 = ----
      3  

$$x_{4} = \frac{5 \pi}{3}$$

Сумма и произведение корней

сумма

    pi   2*pi   4*pi   5*pi
0 + -- + ---- + ---- + ----
    3     3      3      3  

$$\frac{5 \pi}{3} + \left(\left(\left(0 + \frac{\pi}{3}\right) + \frac{2 \pi}{3}\right) + \frac{4 \pi}{3}\right)$$

=

4*pi

$$4 \pi$$

произведение

  pi 2*pi 4*pi 5*pi
1*--*----*----*----
  3   3    3    3  

$$\frac{5 \pi}{3} \frac{4 \pi}{3} \frac{2 \pi}{3} \cdot 1 \frac{\pi}{3}$$

=

     4
40*pi 
------
  81  

$$\frac{40 \pi^{4}}{81}$$

Численный ответ

x1 = 74.3510261349584

x2 = 82.7286065445312

x3 = -55.5014702134197

x4 = 275.412955964705

x5 = 38.7463093942741

x6 = 83.7758040957278

x7 = 33.5103216382911

x8 = -673.348025419412

x9 = -42.9350995990605

x10 = -99.4837673636768

x11 = -96.342174710087

x12 = 26.1799387799149

x13 = -33.5103216382911

x14 = -5.23598775598299

x15 = -85.870199198121

x16 = -76.4454212373516

x17 = -32.4631240870945

x18 = 48.1710873550435

x19 = -90.0589894029074

x20 = 39.7935069454707

x21 = 68.0678408277789

x22 = -30.3687289847013

x23 = 99.4837673636768

x24 = -71.2094334813686

x25 = -52.3598775598299

x26 = 10.471975511966

x27 = 2.0943951023932

x28 = 77.4926187885482

x29 = -79.5870138909414

x30 = -26.1799387799149

x31 = 54.4542726622231

x32 = -48.1710873550435

x33 = -49.2182849062401

x34 = 55.5014702134197

x35 = 98.4365698124802

x36 = -92.1533845053006

x37 = -4.18879020478639

x38 = -98.4365698124802

x39 = -39.7935069454707

x40 = -74.3510261349584

x41 = -27.2271363311115

x42 = -10.471975511966

x43 = -63.8790506229925

x44 = 3803.42150594604

x45 = 269.129770657526

x46 = -24.0855436775217

x47 = 16.7551608191456

x48 = 90.0589894029074

x49 = -35.6047167406843

x50 = 96.342174710087

x51 = 63.8790506229925

x52 = 8.37758040957278

x53 = 24.0855436775217

x54 = -13.6135681655558

x55 = 19.8967534727354

x56 = 61.7846555205993

x57 = 85.870199198121

x58 = -61.7846555205993

x59 = 17.8023583703422

x60 = -19.8967534727354

x61 = -41.8879020478639

x62 = 60.7374579694027

x63 = 41.8879020478639

x64 = -54.4542726622231

x65 = -46.0766922526503

x66 = 52.3598775598299

x67 = -77.4926187885482

x68 = 70.162235930172

x69 = -2.0943951023932

x70 = 4.18879020478639

x71 = 76.4454212373516

x72 = -70.162235930172

x73 = -57.5958653158129

x74 = -11.5191730631626

x75 = 32.4631240870945

x76 = -17.8023583703422

x77 = -68.0678408277789

x78 = 30.3687289847013

x79 = 11.5191730631626

x80 = -93.2005820564972

x81 = 92.1533845053006

x82 = 46.0766922526503

x83 = -83.7758040957278

x84 = -60.7374579694027

График