cos(3*x-pi/2)=1/2 (уравнение)

Найду корень уравнения: cos(3*x-pi/2)=1/2

Вы ввели

   /      pi\      
cos|3*x - --| = 1/2
   \      2 /      

$$\cos{\left(3 x - \frac{\pi}{2} \right)} = \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(3 x - \frac{\pi}{2} \right)} = \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$3 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Или
$$3 x = 2 \pi n + \frac{\pi}{6}$$
$$3 x = 2 \pi n + \frac{5 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$3$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{3} + \frac{\pi}{18}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{5 \pi}{18}$$

Быстрый ответ

     pi
x1 = --
     18

$$x_{1} = \frac{\pi}{18}$$

     5*pi
x2 = ----
      18 

$$x_{2} = \frac{5 \pi}{18}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    18    18 

$$\left(0 + \frac{\pi}{18}\right) + \frac{5 \pi}{18}$$

=

pi
--
3 

$$\frac{\pi}{3}$$

произведение

  pi 5*pi
1*--*----
  18  18 

$$\frac{5 \pi}{18} \cdot 1 \frac{\pi}{18}$$

=

    2
5*pi 
-----
 324 

$$\frac{5 \pi^{2}}{324}$$

Численный ответ

x1 = -47.9965544298441

x2 = -9.59931088596881

x3 = -6.10865238198015

x4 = -99.6583002888762

x5 = -70.3367688553715

x6 = -29.1469985083053

x7 = -72.4311639577647

x8 = 30.1941960595019

x9 = -59.8647933434055

x10 = -66.1479786505851

x11 = -83.6012711705284

x12 = -57.7703982410123

x13 = -34.7320521146872

x14 = 44.1568300754565

x15 = -64.0535835481919

x16 = -1.22173047639603

x17 = -4.01425727958696

x18 = -53.5816080362259

x19 = 94.4223125328932

x20 = -24.9582083035189

x21 = -87.7900613753148

x22 = -22.165681500328

x23 = 50.4400153826361

x24 = -96.1676417848876

x25 = 17.6278254451427

x26 = 83.9503370209273

x27 = 13.4390352403563

x28 = -17.9768912955416

x29 = -26.3544717051144

x30 = 76.2708883121522

x31 = 101.403629540871

x32 = -20.0712863979348

x33 = -73.1292956585624

x34 = 86.0447321233204

x35 = 60.9119908946021

x36 = 28.0998009571087

x37 = -77.3180858633488

x38 = 37.873644768277

x39 = -39.6189740202713

x40 = -13.7881010907552

x41 = -43.8077642250577

x42 = -22.8638132011257

x43 = -45.9021593274509

x44 = 48.3456202802429

x45 = 69.9877030049726

x46 = 23.9110107523223

x47 = 81.8559419185341

x48 = 8.55211333477222

x49 = -41.7133691226645

x50 = -11.693705988362

x51 = 21.8166156499291

x52 = 10.6465084371654

x53 = 63.704517697793

x54 = -68.2423737529783

x55 = 4.36332312998582

x56 = 19.7222205475359

x57 = 54.6288055874225

x58 = -37.5245789178781

x59 = 6.45771823237902

x60 = -94.0732466824944

x61 = 32.2885911618951

x62 = 34.3829862642883

x63 = 52.5344104850293

x64 = -52.1853446346305

x65 = -61.9591884457987

x66 = -97.563905186483

x67 = 42.0624349730633

x68 = 74.176493209759

x69 = 67.8933079025794

x70 = -15.8824961931484

x71 = 90.2335223281068

x72 = -89.884456477708

x73 = -76.6199541625511

x74 = 61.6101225953998

x75 = 103.498024643264

x76 = -1.91986217719376

x77 = -50.0909495322373

x78 = -85.6956662729216

x79 = 96.5167076352864

x80 = 39.9680398706701

x81 = 72.0820981073658

x82 = 88.1391272257137

x83 = -55.6760031386191

x84 = 2.26892802759263

x85 = 26.0054058547155

x86 = 65.7989128001862

x87 = 0.174532925199433

x88 = -35.4301838154849

x89 = -91.9788515801012

x90 = 92.3279174305

x91 = 46.2512251778497

x92 = 98.6111027376796

x93 = 78.3652834145454

x94 = -79.412480965742

x95 = -24.2600766027212

x96 = -81.5068760681352

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