cos(pi/2-3x/2)=sqrt(3)/2 (уравнение)

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

Вы ввели

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

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

Подробное решение

Дано уравнение
$$\cos{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)} = \frac{\sqrt{3}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$\frac{3 x}{2} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$\frac{3 x}{2} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi$$
Или
$$\frac{3 x}{2} = 2 \pi n + \frac{\pi}{3}$$
$$\frac{3 x}{2} = 2 \pi n + \frac{2 \pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$\frac{3}{2}$$
получим ответ:
$$x_{1} = \frac{4 \pi n}{3} + \frac{2 \pi}{9}$$
$$x_{2} = \frac{4 \pi n}{3} + \frac{4 \pi}{9}$$

Быстрый ответ

     2*pi
x1 = ----
      9  

$$x_{1} = \frac{2 \pi}{9}$$

     4*pi
x2 = ----
      9  

$$x_{2} = \frac{4 \pi}{9}$$

Сумма и произведение корней

сумма

    2*pi   4*pi
0 + ---- + ----
     9      9  

$$\left(0 + \frac{2 \pi}{9}\right) + \frac{4 \pi}{9}$$

=

2*pi
----
 3  

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

произведение

  2*pi 4*pi
1*----*----
   9    9  

$$\frac{4 \pi}{9} \cdot 1 \cdot \frac{2 \pi}{9}$$

=

    2
8*pi 
-----
  81 

$$\frac{8 \pi^{2}}{81}$$

Численный ответ

x1 = 97.7384381116825

x2 = -2.79252680319093

x3 = 772.133661082291

x4 = 34.2084533390889

x5 = 55.1524043630208

x6 = 71.9075651821664

x7 = -7.67944870877505

x8 = -23.7364778271229

x9 = 97.0403064108847

x10 = -1976.41084495838

x11 = 85.1720674973233

x12 = 80.2851455917392

x13 = 42.5860337486616

x14 = -82.3795406941324

x15 = -41.1897703470662

x16 = 21.6420827247297

x17 = 5.58505360638185

x18 = 43.2841654494594

x19 = 59.3411945678072

x20 = 0.698131700797732

x21 = -45.3785605518526

x22 = -91.4552528045029

x23 = -40.4916386462684

x24 = -32.1140582366957

x25 = 84.4739357965256

x26 = -74.0019602845596

x27 = 68.4169066781777

x28 = 1.39626340159546

x29 = -6.98131700797732

x30 = 47.4729556542458

x31 = 9.07571211037051

x32 = -57.9449311662117

x33 = -19.5476876223365

x34 = -90.7571211037051

x35 = -99.8328332140756

x36 = 88.6627260013119

x37 = -69.8131700797732

x38 = -95.6440430092893

x39 = 50.9636141582344

x40 = 34.9065850398866

x41 = 93.5496479068961

x42 = 64.2281164733913

x43 = -53.0580092606276

x44 = -44.6804288510548

x45 = -78.190750489346

x46 = 13.2645023151569

x47 = -27.9252680319093

x48 = 25.8308729295161

x49 = 18.151424220741

x50 = 38.3972435438752

x51 = -53.7561409614254

x52 = -36.3028484414821

x53 = -83.0776723949301

x54 = 13.9626340159546

x55 = 51.6617458590322

x56 = 76.0963553869528

x57 = -49.567350756639

x58 = -3.49065850398866

x59 = 612.959633300409

x60 = -99.1347015132779

x61 = -86.5683308989187

x62 = 17.4532925199433

x63 = 22.3402144255274

x64 = -32.8121899374934

x65 = -62.1337213709981

x66 = 9.77384381116824

x67 = 92.8515162060983

x68 = 39.095375244673

x69 = 46.774823953448

x70 = 691.848515490552

x71 = -16.0570291183478

x72 = -48.8692190558412

x73 = 60.0393262686049

x74 = 80.9832772925369

x75 = 55.8505360638185

x76 = -11.8682389135614

x77 = 30.0196631343025

x78 = -78.8888821901437

x79 = -65.6243798749868

x80 = 4.88692190558412

x81 = 101.229096615671

x82 = 89.3608577021097

x83 = -74.7000919853573

x84 = -28.623399732707

x85 = -94.9459113084915

x86 = -87.2664625997165

x87 = -37.0009801422798

График