2sin2x=-1 (уравнение)

Найду корень уравнения: 2sin2x=-1

Вы ввели

2*sin(2*x) = -1

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

Подробное решение

Дано уравнение
$$2 \sin{\left(2 x \right)} = -1$$
- 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 = 42.6733002112614

x3 = 37.4373124552784

x4 = -20.1585528605345

x5 = 36.3901149040818

x6 = -7.59218224617533

x7 = -64.1408500107916

x8 = 18.5877565337396

x9 = -44.2440965380563

x10 = 78.2780169519457

x11 = -86.1319985859202

x12 = -67.2824426643814

x13 = 43.720497762458

x14 = -79.8488132787406

x15 = -51.5744793964324

x16 = -9.68657734856853

x17 = 65.7116463375865

x18 = 34.2957198016886

x19 = 6.02138591938044

x20 = 56.2868683768171

x21 = 74.0892267471593

x22 = -89.27359123951

x23 = 12.30457122656

x24 = 80.3724120543389

x25 = -95.5567765466895

x26 = -28.5361332701073

x27 = 59.4284610304069

x28 = -17.0169602069447

x29 = -1.30899693899575

x30 = -94.5095789954929

x31 = -15.9697626557481

x32 = 87.7027949127151

x33 = -53.6688744988256

x34 = 14.3989663289532

x35 = -45.2912940892529

x36 = -88.2263936883134

x37 = 52.0980781720307

x38 = 81.4196096055355

x39 = 23.8237442897226

x40 = -22.2529479629277

x41 = 20.6821516361328

x42 = -1926.05809603834

x43 = 15.4461638801498

x44 = 50.0036830696375

x45 = -23.3001455141243

x46 = 62.5700536839967

x47 = -66.2352451131848

x48 = 71.9948316447661

x49 = -31.6777259236971

x50 = -75.6600230739542

x51 = 90.8443875663049

x52 = -72.5184304203644

x53 = -13.8753675533549

x54 = -91.3679863419031

x55 = 93.9859802198946

x56 = -50.5272818452358

x57 = 96.0803753222878

x58 = -0.261799387799149

x59 = 1.83259571459405

x60 = -59.9520598060052

x61 = 40.5789051088682

x62 = 127.496301858186

x63 = -81.9432083811338

x64 = 67.8060414399797

x65 = -37.9609112308767

x66 = 84.5612022591253

x67 = 45.8148928648512

x68 = 89.7971900151083

x69 = -73.565627971561

x70 = -42.1497014356631

x71 = 100.269165527074

x72 = 28.012534494509

x73 = 58.3812634792103

x74 = -6.54498469497874

x75 = 64.6644487863899

x76 = 21.7293491873294

x77 = -57.857664703612

x78 = -97.6511716490827

x79 = 8.11578102177363

x80 = 30.1069295969022

x81 = -29.5833308213039

x82 = 86.6555973615185

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