sin4x=√3/2 (уравнение)

Найду корень уравнения: sin4x=√3/2

Вы ввели

             ___
           \/ 3 
sin(4*x) = -----
             2  

$$\sin{\left(4 x \right)} = \frac{\sqrt{3}}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(4 x \right)} = \frac{\sqrt{3}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi$$
Или
$$4 x = 2 \pi n + \frac{\pi}{3}$$
$$4 x = 2 \pi n + \frac{2 \pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{2} + \frac{\pi}{6}$$

Быстрый ответ

     pi
x1 = --
     12

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

     pi
x2 = --
     6 

$$x_{2} = \frac{\pi}{6}$$

Сумма и произведение корней

сумма

    pi   pi
0 + -- + --
    12   6 

$$\left(0 + \frac{\pi}{12}\right) + \frac{\pi}{6}$$

=

pi
--
4 

$$\frac{\pi}{4}$$

произведение

  pi pi
1*--*--
  12 6 

$$\frac{\pi}{6} \cdot 1 \frac{\pi}{12}$$

=

  2
pi 
---
 72

$$\frac{\pi^{2}}{72}$$

Численный ответ

x1 = -35.8665161284835

x2 = -92.1533845053006

x3 = 36.3901149040818

x4 = -43.720497762458

x5 = 91.6297857297023

x6 = -7.59218224617533

x7 = 90.0589894029074

x8 = -79.8488132787406

x9 = 47.6474885794452

x10 = -51.5744793964324

x11 = -93.7241808320955

x12 = 82.2050077689329

x13 = -81.4196096055355

x14 = 74.0892267471593

x15 = -15.4461638801498

x16 = 2.0943951023932

x17 = 83.7758040957278

x18 = -34.0339204138894

x19 = 5.23598775598299

x20 = 80.3724120543389

x21 = -95.5567765466895

x22 = -4.18879020478639

x23 = -31.1541271480988

x24 = -59.4284610304069

x25 = 0.261799387799149

x26 = -12.0427718387609

x27 = -56.025068989018

x28 = -27.7507351067098

x29 = -18.3259571459405

x30 = 53.9306738866248

x31 = 14.3989663289532

x32 = 91.3679863419031

x33 = 52.0980781720307

x34 = -78.0162175641465

x35 = 3.66519142918809

x36 = 68.0678408277789

x37 = 171.478599008443

x38 = 64.9262481741891

x39 = 22.2529479629277

x40 = -19.8967534727354

x41 = -87.7027949127151

x42 = -71.733032256967

x43 = -85.870199198121

x44 = -70.162235930172

x45 = 17.8023583703422

x46 = 66.2352451131848

x47 = 25.6563400043166

x48 = -2167.43713158916

x49 = 86.9173967493176

x50 = 46.0766922526503

x51 = 44.2440965380563

x52 = 69.3768377667746

x53 = -40.317105721069

x54 = 61.7846555205993

x55 = -62.3082542961976

x56 = 60.2138591938044

x57 = -56.2868683768171

x58 = -13.8753675533549

x59 = 6.54498469497874

x60 = -26.1799387799149

x61 = 96.0803753222878

x62 = 97.9129710368819

x63 = 94.5095789954929

x64 = -5.75958653158129

x65 = 50.5272818452358

x66 = -21.7293491873294

x67 = -100.007366139275

x68 = 9.94837673636768

x69 = -65.7116463375865

x70 = 31.9395253114962

x71 = 75.9218224617533

x72 = -73.565627971561

x73 = -49.7418836818384

x74 = -63.8790506229925

x75 = 38.2227106186758

x76 = -48.1710873550435

x77 = 39.7935069454707

x78 = 24.0855436775217

x79 = 58.3812634792103

x80 = 28.5361332701073

x81 = 88.2263936883134

x82 = -57.857664703612

x83 = -84.2994028713261

x84 = 72.5184304203644

x85 = 8.11578102177363

x86 = -41.8879020478639

x87 = -37.4373124552784

x88 = 16.2315620435473

x89 = 30.1069295969022

x90 = -29.5833308213039

График