cos(2*x-pi/6)=-1 (уравнение)

Найду корень уравнения: cos(2*x-pi/6)=-1

Вы ввели

   /      pi\     
cos|2*x - --| = -1
   \      6 /     

$$\cos{\left(2 x - \frac{\pi}{6} \right)} = -1$$

Подробное решение

Дано уравнение
$$\cos{\left(2 x - \frac{\pi}{6} \right)} = -1$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$2 x + \frac{\pi}{3} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$2 x + \frac{\pi}{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Или
$$2 x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{2}$$
$$2 x + \frac{\pi}{3} = 2 \pi n + \frac{3 \pi}{2}$$
, где n - любое целое число
Перенесём
$$\frac{\pi}{3}$$
в правую часть ур-ния
с противоположным знаком, итого:
$$2 x = 2 \pi n - \frac{5 \pi}{6}$$
$$2 x = 2 \pi n + \frac{7 \pi}{6}$$
Разделим обе части полученного ур-ния на
$$2$$
получим ответ:
$$x_{1} = \pi n - \frac{5 \pi}{12}$$
$$x_{2} = \pi n + \frac{7 \pi}{12}$$

Быстрый ответ

     -5*pi
x1 = -----
       12 

$$x_{1} = - \frac{5 \pi}{12}$$

     7*pi
x2 = ----
      12 

$$x_{2} = \frac{7 \pi}{12}$$

Сумма и произведение корней

сумма

    5*pi   7*pi
0 - ---- + ----
     12     12 

$$\left(- \frac{5 \pi}{12} + 0\right) + \frac{7 \pi}{12}$$

=

pi
--
6 

$$\frac{\pi}{6}$$

произведение

  -5*pi 7*pi
1*-----*----
    12   12 

$$\frac{7 \pi}{12} \cdot 1 \left(- \frac{5 \pi}{12}\right)$$

=

      2
-35*pi 
-------
  144  

$$- \frac{35 \pi^{2}}{144}$$

Численный ответ

x1 = -89.2735914552943

x2 = -64.1408499069291

x3 = -92.4151836784452

x4 = 33.248522502985

x5 = -60.9992576050299

x6 = -39.0081087067327

x7 = -79.8488134219953

x8 = 96.0803749965022

x9 = 61.5228562124743

x10 = 99.2219682241048

x11 = 23.823744411737

x12 = -20.1585527474046

x13 = -29.5833309181307

x14 = -13.8753674921739

x15 = -48.4328865231738

x16 = -23.3001457224198

x17 = 14.3989662408997

x18 = 26.9653369615449

x19 = -73.56562807693

x20 = -7.59218233864531

x21 = -82.9904061792384

x22 = 92.938782418861

x23 = 1.83259583221116

x24 = 45.8148929911776

x25 = -32.7249232196457

x26 = -39.0081090306488

x27 = 55.23967107687

x28 = 83.5140042317425

x29 = 77.2308196505795

x30 = 55.2396707188534

x31 = 39.5317077865533

x32 = 4.97418811860037

x33 = -95.5567766562468

x34 = 26.9653366934544

x35 = 61.5228563638098

x36 = 83.514004940994

x37 = -76.7072203688128

x38 = -67.2824428777351

x39 = 20.6821514331505

x40 = -17.0169601489251

x41 = -76.707220622515

x42 = 67.8060415705369

x43 = -10.7337746452799

x44 = 4.97418840722781

x45 = -10.7337749592817

x46 = -54.7160717941586

x47 = 58.381263400178

x48 = 64.664448588967

x49 = 48.956485268453

x50 = -26.4417379456388

x51 = -89.2735913181908

x52 = -54.7160720700577

x53 = -32.7249235155804

x54 = 42.6733000110285

x55 = 92.9387826113621

x56 = -45.2912943001099

x57 = -45.291294636148

x58 = -42.1497013271158

x59 = -60.9992572660328

x60 = 30.1069302014112

x61 = 70.947634063845

x62 = 99.2219678439199

x63 = 8.1157813951878

x64 = -79.8488131572011

x65 = -70.4240351007762

x66 = 48.9564855138247

x67 = -51.5744794975584

x68 = 11.2573735986602

x69 = -35.866515952878

x70 = 80.3724119799169

x71 = -98.6983691727318

x72 = -98.698368943603

x73 = -82.9904058266802

x74 = 86.6555971669651

x75 = -86.1319984868519

x76 = -57.857664557111

x77 = 89.7971901498186

x78 = 11.2573739289326

x79 = 77.2308192808357

x80 = 36.3901148205066

x81 = -17.0169604561029

x82 = 33.248522158086

x83 = -1.30899714466595

x84 = -4.45058936817196

x85 = 17.5405592092257

x86 = 70.9476338435903

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