2sin(pi+2x)=1 (уравнение)

Найду корень уравнения: 2sin(pi+2x)=1

Вы ввели

2*sin(pi + 2*x) = 1

$$2 \sin{\left(2 x + \pi \right)} = 1$$

Подробное решение

Дано уравнение
$$2 \sin{\left(2 x + \pi \right)} = 1$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на -2

Ур-ние превратится в
$$\sin{\left(2 x \right)} = - \frac{1}{2}$$
Это ур-ние преобразуется в
$$2 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Или
$$2 x = 2 \pi n - \frac{\pi}{6}$$
$$2 x = 2 \pi n + \frac{7 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$2$$
получим ответ:
$$x_{1} = \pi n - \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{7 \pi}{12}$$

Быстрый ответ

     -pi 
x1 = ----
      12 

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

     7*pi
x2 = ----
      12 

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

Сумма и произведение корней

сумма

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

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

=

pi
--
2 

$$\frac{\pi}{2}$$

произведение

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

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

=

     2
-7*pi 
------
 144  

$$- \frac{7 \pi^{2}}{144}$$

Численный ответ

x1 = -35.8665161284835

x2 = 74.0892267471593

x3 = -64.1408500107916

x4 = -72.5184304203644

x5 = -66.2352451131848

x6 = -86.1319985859202

x7 = -57.857664703612

x8 = 1.83259571459405

x9 = -97.6511716490827

x10 = -50.5272818452358

x11 = 90.8443875663049

x12 = 34.2957198016886

x13 = -17.0169602069447

x14 = 40.5789051088682

x15 = 65.7116463375865

x16 = 71.9948316447661

x17 = -59.9520598060052

x18 = -45.2912940892529

x19 = 59.4284610304069

x20 = -88.2263936883134

x21 = -73.565627971561

x22 = -89.27359123951

x23 = -1.30899693899575

x24 = 84.5612022591253

x25 = 89.7971900151083

x26 = 36.3901149040818

x27 = 37.4373124552784

x28 = 87.7027949127151

x29 = -42.1497014356631

x30 = 20.6821516361328

x31 = 86.6555973615185

x32 = -81.9432083811338

x33 = 100.269165527074

x34 = -13.8753675533549

x35 = -95.5567765466895

x36 = 56.2868683768171

x37 = -51.5744793964324

x38 = 67.8060414399797

x39 = 28.012534494509

x40 = -31.6777259236971

x41 = 21.7293491873294

x42 = 18.5877565337396

x43 = 14.3989663289532

x44 = 62.5700536839967

x45 = 50.0036830696375

x46 = -79.8488132787406

x47 = -37.9609112308767

x48 = -29.5833308213039

x49 = 93.9859802198946

x50 = 12.30457122656

x51 = -1926.05809603834

x52 = -53.6688744988256

x53 = -67.2824426643814

x54 = 45.8148928648512

x55 = -94.5095789954929

x56 = -0.261799387799149

x57 = 8.11578102177363

x58 = -75.6600230739542

x59 = 78.2780169519457

x60 = 30.1069295969022

x61 = 23.8237442897226

x62 = -22.2529479629277

x63 = 43.720497762458

x64 = -6.54498469497874

x65 = -44.2440965380563

x66 = 96.0803753222878

x67 = 6.02138591938044

x68 = -28.5361332701073

x69 = -91.3679863419031

x70 = 42.6733002112614

x71 = 80.3724120543389

x72 = 64.6644487863899

x73 = 81.4196096055355

x74 = -9.68657734856853

x75 = 127.496301858186

x76 = 52.0980781720307

x77 = -7.59218224617533

x78 = -20.1585528605345

x79 = 58.3812634792103

x80 = 15.4461638801498

x81 = -23.3001455141243

x82 = -15.9697626557481

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