2sin(2x)=1 (уравнение)

Найду корень уравнения: 2sin(2x)=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
$$2 x = 2 \pi n + \frac{\pi}{6}$$
$$2 x = 2 \pi n + \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$2$$
получим ответ:
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$

Быстрый ответ

     pi
x1 = --
     12

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

     5*pi
x2 = ----
      12 

$$x_{2} = \frac{5 \pi}{12}$$

Сумма и произведение корней

сумма

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

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

=

pi
--
2 

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

произведение

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

$$\frac{5 \pi}{12} \cdot 1 \frac{\pi}{12}$$

=

    2
5*pi 
-----
 144 

$$\frac{5 \pi^{2}}{144}$$

Численный ответ

x1 = 0.261799387799149

x2 = -100.269165527074

x3 = -8.11578102177363

x4 = -52.0980781720307

x5 = -9.16297857297023

x6 = 72.5184304203644

x7 = 6.54498469497874

x8 = 31.6777259236971

x9 = -1.83259571459405

x10 = -83.5140047079287

x11 = -86.6555973615185

x12 = 70.4240353179712

x13 = -34.2957198016886

x14 = -81.4196096055355

x15 = -6.02138591938044

x16 = 78.801615727544

x17 = -78.2780169519457

x18 = -17.540558982543

x19 = -67.8060414399797

x20 = 75.6600230739542

x21 = 94.5095789954929

x22 = 35.8665161284835

x23 = 4.45058959258554

x24 = -55.2396708256205

x25 = 59.9520598060052

x26 = -28.012534494509

x27 = 48.4328867428426

x28 = -59.4284610304069

x29 = -21.7293491873294

x30 = -694.030177055545

x31 = -89.7971900151083

x32 = -87.7027949127151

x33 = -58.3812634792103

x34 = -23.8237442897226

x35 = -30.1069295969022

x36 = -14.3989663289532

x37 = 53.6688744988256

x38 = 86.1319985859202

x39 = 73.565627971561

x40 = 79.8488132787406

x41 = 29.5833308213039

x42 = 26.4417381677141

x43 = 42.1497014356631

x44 = -37.4373124552784

x45 = 12.8281700021583

x46 = 28.5361332701073

x47 = -64.6644487863899

x48 = 37.9609112308767

x49 = 45.2912940892529

x50 = -93.9859802198946

x51 = -53.1452757232273

x52 = -39.5317075576716

x53 = -36.3901149040818

x54 = 57.857664703612

x55 = -75.1364242983559

x56 = 64.1408500107916

x57 = -56.2868683768171

x58 = -61.5228561328001

x59 = 51.5744793964324

x60 = -43.720497762458

x61 = 100.792764302673

x62 = 81.9432083811338

x63 = -42.6733002112614

x64 = 15.9697626557481

x65 = 7.59218224617533

x66 = 56.8104671524154

x67 = 20.1585528605345

x68 = -97.1275728734844

x69 = 44.2440965380563

x70 = -15.4461638801498

x71 = 9.68657734856853

x72 = -96.0803753222878

x73 = 13.8753675533549

x74 = -12.30457122656

x75 = 95.5567765466895

x76 = 92.4151838930998

x77 = -71.9948316447661

x78 = 50.5272818452358

x79 = -65.7116463375865

x80 = -31.1541271480988

x81 = 22.2529479629277

x82 = 88.2263936883134

x83 = -74.0892267471593

x84 = 34.8193185772869

x85 = -40.5789051088682

x86 = -45.8148928648512

x87 = 66.2352451131848

x88 = -80.3724120543389

x89 = -50.0036830696375

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