sin(2x)=√2/2 (уравнение)

Найду корень уравнения: sin(2x)=√2/2

Вы ввели

             ___
           \/ 2 
sin(2*x) = -----
             2  

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

Подробное решение

Дано уравнение
$$\sin{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Или
$$2 x = 2 \pi n + \frac{\pi}{4}$$
$$2 x = 2 \pi n + \frac{3 \pi}{4}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$2$$
получим ответ:
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$

Быстрый ответ

     pi
x1 = --
     8 

$$x_{1} = \frac{\pi}{8}$$

     3*pi
x2 = ----
      8  

$$x_{2} = \frac{3 \pi}{8}$$

Сумма и произведение корней

сумма

    pi   3*pi
0 + -- + ----
    8     8  

$$\left(0 + \frac{\pi}{8}\right) + \frac{3 \pi}{8}$$

=

pi
--
2 

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

произведение

  pi 3*pi
1*--*----
  8   8  

$$\frac{3 \pi}{8} \cdot 1 \frac{\pi}{8}$$

=

    2
3*pi 
-----
  64 

$$\frac{3 \pi^{2}}{64}$$

Численный ответ

x1 = -39.6626072515711

x2 = 4.31968989868597

x3 = -12.1736715326604

x4 = -58.5121631731099

x5 = -56.1559686829176

x6 = 79.717913584841

x7 = -5901.87449885013

x8 = -17.6714586764426

x9 = -36.5210145979813

x10 = 73.4347282776614

x11 = -65.5807466436869

x12 = -21.5984494934298

x13 = -31.0232274541992

x14 = 20.0276531666349

x15 = -61.6537558266997

x16 = 38.0918109247762

x17 = 53.7997741927252

x18 = 35.7356164345839

x19 = 48.3019870489431

x20 = 92.2842841992002

x21 = -55.3705705195201

x22 = 75.7909227678538

x23 = 26.3108384738145

x24 = 13.7444678594553

x25 = 82.0741080750334

x26 = 42.0188017417635

x27 = 60.0829594999048

x28 = -5.89048622548086

x29 = 6.67588438887831

x30 = 66.3661448070844

x31 = -27.8816348006094

x32 = 64.009950316892

x33 = -23.9546439836222

x34 = -78.1471172580461

x35 = -30.2378292908018

x36 = 97.7820713429823

x37 = -49.872783375738

x38 = -15.3152641862502

x39 = 27323.6094055155

x40 = 22.3838476568273

x41 = 4622.46089067568

x42 = 94.6404786893925

x43 = 100.923663996572

x44 = 95.42587685279

x45 = 104.850654813559

x46 = 0.392699081698724

x47 = -9.03207887907065

x48 = -96.2112750161874

x49 = 29.4524311274043

x50 = -43.5895980685584

x51 = 16.1006623496477

x52 = 57.7267650097125

x53 = -83.6449044018282

x54 = 7.46128255227576

x55 = -87.5718952188155

x56 = 28.6670329640069

x57 = -53.0143760293278

x58 = -89.9280897090078

x59 = -81.2887099116359

x60 = 12.9590696960579

x61 = 86.0010988920206

x62 = 56.941366846315

x63 = -67.9369411338793

x64 = 50.6581815391354

x65 = 9.8174770424681

x66 = -99.3528676697772

x67 = -8.24668071567321

x68 = 88.3572933822129

x69 = -80.5033117482384

x70 = 70.2931356240716

x71 = -75.0055246044563

x72 = 51.4435797025329

x73 = -1.96349540849362

x74 = 34.9502182711865

x75 = -14.5298660228528

x76 = -52.2289778659303

x77 = -71.8639319508665

x78 = 31.8086256175967

x79 = 44.3749962319558

x80 = -93.8550805259951

x81 = 78.9325154214436

x82 = -45.9457925587507

x83 = 89.1426915456104

x84 = -74.2201264410589

x85 = -59.2975613365073

x86 = -34.164820107789

x87 = 72.649330114264

x88 = -37.3064127613788

x89 = -100.138265833175

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