sin5x=√3/2 (уравнение)

Найду корень уравнения: sin5x=√3/2

Вы ввели

             ___
           \/ 3 
sin(5*x) = -----
             2  

$$\sin{\left(5 x \right)} = \frac{\sqrt{3}}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(5 x \right)} = \frac{\sqrt{3}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$5 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi$$
Или
$$5 x = 2 \pi n + \frac{\pi}{3}$$
$$5 x = 2 \pi n + \frac{2 \pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$5$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{5} + \frac{\pi}{15}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{2 \pi}{15}$$

Быстрый ответ

     pi
x1 = --
     15

$$x_{1} = \frac{\pi}{15}$$

     2*pi
x2 = ----
      15 

$$x_{2} = \frac{2 \pi}{15}$$

Сумма и произведение корней

сумма

    pi   2*pi
0 + -- + ----
    15    15 

$$\left(0 + \frac{\pi}{15}\right) + \frac{2 \pi}{15}$$

=

pi
--
5 

$$\frac{\pi}{5}$$

произведение

  pi 2*pi
1*--*----
  15  15 

$$\frac{2 \pi}{15} \cdot 1 \frac{\pi}{15}$$

=

    2
2*pi 
-----
 225 

$$\frac{2 \pi^{2}}{225}$$

Численный ответ

x1 = 16.7551608191456

x2 = -59.8996999284454

x3 = -67.6489618073002

x4 = 48.1710873550435

x5 = -56.1297887441376

x6 = -47.5427688243255

x7 = 65.7640062151463

x8 = 88.3834733209928

x9 = 82.1002880138133

x10 = -12.1474915938805

x11 = -42.5162205785819

x12 = -33.7197611485304

x13 = 76.8643002578303

x14 = -43.7728576400178

x15 = 34.1386401690091

x16 = 50.474921967676

x17 = 41.8879020478639

x18 = -23.6666646570431

x19 = -49.8466034369581

x20 = -83.7758040957278

x21 = 92.1533845053006

x22 = -46.0766922526503

x23 = 44.1917366604964

x24 = 85.870199198121

x25 = 54.2448331519838

x26 = 98.2271303022409

x27 = 1219.14738910308

x28 = -69.9527964199327

x29 = -53.6165146212658

x30 = 147.235975698242

x31 = -40.00294645571

x32 = 30.3687289847013

x33 = -24.7138622082397

x34 = 72.0471915223259

x35 = 38.1179908635562

x36 = 88.1740338107535

x37 = 31.8348055563766

x38 = 14.2418866962737

x39 = -87.7551547902749

x40 = -86.498517728839

x41 = -3.56047167406843

x42 = 7.95870138909414

x43 = 40.4218254761887

x44 = -57.5958653158129

x45 = -66.182885235625

x46 = 20.315632493214

x47 = 95.9232956896084

x48 = -29.7404104539834

x49 = -81.262529972856

x50 = 11.7286125734019

x51 = 55.5014702134197

x52 = 74.3510261349584

x53 = -19.687313962496

x54 = 28.0648943720688

x55 = -15.9174027781883

x56 = -37.4896723328382

x57 = -1952.39511445094

x58 = 0.20943951023932

x59 = 60.7374579694027

x60 = -80.0058929114201

x61 = -93.8289005872152

x62 = 18.0117978805815

x63 = -117.705004754498

x64 = -97.8082512817622

x65 = 11.5191730631626

x66 = 99.4837673636768

x67 = -32.2536845768552

x68 = 84.4041226264458

x69 = -5.86430628670095

x70 = 51.9409985393512

x71 = 6.49262481741891

x72 = -13.6135681655558

x73 = 24.2949831877611

x74 = 10.2625360017267

x75 = -90.0589894029074

x76 = -73.7227076042405

x77 = -77.7020582987875

x78 = 64.2979296434711

x79 = 68.2772803380182

x80 = -53.8259541315051

x81 = -86.2890782185997

x82 = -65.1356876844284

x83 = -36.023595761163

x84 = -22.2005880853679

x85 = 58.2241838465308

x86 = -100.112085894395

x87 = 21.7817090648892

x88 = 35.395277230445

x89 = 94.4572191179331

x90 = -9.84365698124802

x91 = -76.2359817271123

x92 = -91.5250659745826

x93 = -25.9704992696756

x94 = 61.9940950308386

x95 = -2.0943951023932

x96 = 75.8171027066337

График