|sin(x)|=1/2 (уравнение)

Найду корень уравнения: |sin(x)|=1/2

Вы ввели

|sin(x)| = 1/2

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

Подробное решение

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

Быстрый ответ

     -pi 
x1 = ----
      6  

$$x_{1} = - \frac{\pi}{6}$$

     pi
x2 = --
     6 

$$x_{2} = \frac{\pi}{6}$$

     5*pi
x3 = ----
      6  

$$x_{3} = \frac{5 \pi}{6}$$

     7*pi
x4 = ----
      6  

$$x_{4} = \frac{7 \pi}{6}$$

Сумма и произведение корней

сумма

    pi   pi   5*pi   7*pi
0 - -- + -- + ---- + ----
    6    6     6      6  

$$\left(\left(\left(- \frac{\pi}{6} + 0\right) + \frac{\pi}{6}\right) + \frac{5 \pi}{6}\right) + \frac{7 \pi}{6}$$

=

2*pi

$$2 \pi$$

произведение

  -pi  pi 5*pi 7*pi
1*----*--*----*----
   6   6   6    6  

$$\frac{7 \pi}{6} \frac{5 \pi}{6} \frac{\pi}{6} \cdot 1 \left(- \frac{\pi}{6}\right)$$

=

      4
-35*pi 
-------
  1296 

$$- \frac{35 \pi^{4}}{1296}$$

Численный ответ

x1 = -19.3731546971371

x2 = -68.5914396033772

x3 = 40.317105721069

x4 = 5.75958653158129

x5 = 91.6297857297023

x6 = -85.3466004225227

x7 = -24.60914245312

x8 = -3.66519142918809

x9 = 22.5147473507269

x10 = 88.4881930761125

x11 = 47.6474885794452

x12 = 84.2994028713261

x13 = -93.7241808320955

x14 = 82.2050077689329

x15 = 90.5825881785057

x16 = 0.523598775598299

x17 = 46.6002910282486

x18 = -53.9306738866248

x19 = -34.0339204138894

x20 = -46.6002910282486

x21 = -12.0427718387609

x22 = -21.4675497995303

x23 = -56.025068989018

x24 = -27.7507351067098

x25 = -31.9395253114962

x26 = -18.3259571459405

x27 = 53.9306738866248

x28 = -137.706477982353

x29 = 62.3082542961976

x30 = -60.2138591938044

x31 = 18.3259571459405

x32 = 27.7507351067098

x33 = -78.0162175641465

x34 = -16.2315620435473

x35 = 3.66519142918809

x36 = 44.5058959258554

x37 = -43.4586983746588

x38 = -87.4409955249159

x39 = -2.61799387799149

x40 = -69.6386371545737

x41 = 34.0339204138894

x42 = 69.6386371545737

x43 = 2.61799387799149

x44 = -71.733032256967

x45 = -66.497044500984

x46 = 78.0162175641465

x47 = 66.497044500984

x48 = -91.6297857297023

x49 = 25.6563400043166

x50 = -25.6563400043166

x51 = -40.317105721069

x52 = -97.9129710368819

x53 = -63.3554518473942

x54 = -82.2050077689329

x55 = -62.3082542961976

x56 = -65.4498469497874

x57 = -41.3643032722656

x58 = 60.2138591938044

x59 = 49.7418836818384

x60 = 100.007366139275

x61 = -47.6474885794452

x62 = 97.9129710368819

x63 = -5.75958653158129

x64 = 68.5914396033772

x65 = 56.025068989018

x66 = -100.007366139275

x67 = 9.94837673636768

x68 = 31.9395253114962

x69 = 75.9218224617533

x70 = 71.733032256967

x71 = -49.7418836818384

x72 = 12.0427718387609

x73 = 96.8657734856853

x74 = -38.2227106186758

x75 = -57.0722665402146

x76 = 38.2227106186758

x77 = 93.7241808320955

x78 = -75.9218224617533

x79 = -90.5825881785057

x80 = -382.750704962356

x81 = -84.2994028713261

x82 = 24.60914245312

x83 = 16.2315620435473

x84 = -9.94837673636768

График