cos(3x+2)=0,5 (уравнение)

Найду корень уравнения: cos(3x+2)=0,5

Вы ввели

cos(3*x + 2) = 1/2

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

Подробное решение

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

Быстрый ответ

       2   pi
x1 = - - + --
       3   9 

$$x_{1} = - \frac{2}{3} + \frac{\pi}{9}$$

       2   5*pi
x2 = - - + ----
       3    9  

$$x_{2} = - \frac{2}{3} + \frac{5 \pi}{9}$$

Сумма и произведение корней

сумма

  2   pi     2   5*pi
- - + -- + - - + ----
  3   9      3    9  

$$\left(- \frac{2}{3} + \frac{\pi}{9}\right) + \left(- \frac{2}{3} + \frac{5 \pi}{9}\right)$$

=

  4   2*pi
- - + ----
  3    3  

$$- \frac{4}{3} + \frac{2 \pi}{3}$$

произведение

/  2   pi\ /  2   5*pi\
|- - + --|*|- - + ----|
\  3   9 / \  3    9  /

$$\left(- \frac{2}{3} + \frac{\pi}{9}\right) \left(- \frac{2}{3} + \frac{5 \pi}{9}\right)$$

=

(6 - pi)*(6 - 5*pi)
-------------------
         81        

$$\frac{\left(6 - 5 \pi\right) \left(6 - \pi\right)}{81}$$

Численный ответ

x1 = 60.4198571531349

x2 = 1.77679428612539

x3 = 34.5889842236188

x4 = 47.8534865387757

x5 = 74.3824911690895

x6 = 8.05997959330498

x7 = -65.9419806912546

x8 = 37.3815110268097

x9 = 278.236947802027

x10 = 89.7413885866396

x11 = -0.317600816267801

x12 = 40.8721695307984

x13 = 3.87118938851859

x14 = -15.6764982338179

x15 = 43.6646963339893

x16 = 16.4375600028778

x17 = 29.0039306172369

x18 = -111.320541243107

x19 = -97.3579072271525

x20 = -53.3756100768954

x21 = -35.9223175569521

x22 = -81.9990098096024

x23 = 87.6469934842464

x24 = 54.1366718459553

x25 = 96.0245738938192

x26 = -42.2055028641317

x27 = 42.9665646331916

x28 = 68.0993058619099

x29 = 22.0226136092596

x30 = -11.4877080290315

x31 = -13.5821031314247

x32 = 72.2880960666963

x33 = -29.6391322497725

x34 = -21.9596835409975

x35 = -59.658795384075

x36 = -26.1484737457839

x37 = 32.4945891212256

x38 = -48.4886881713113

x39 = -19.8652884386043

x40 = 78.5712813738759

x41 = 98.1189689962124

x42 = 12.2487697980914

x43 = -46.3942930689181

x44 = -75.7158245024228

x45 = 100.213364098606

x46 = -17.7708933362111

x47 = -73.6214294000296

x48 = -3.11012761945873

x49 = -55.4700051792886

x50 = -2.411995918661

x51 = -88.282195116782

x52 = 52.0422767435621

x53 = -24.0540786433907

x54 = -33.8279224545589

x55 = 30.4001940188324

x56 = -84.0934049119956

x57 = -79.9046147072092

x58 = 66.0049107595167

x59 = 19.9282185068664

x60 = -6.60078612344739

x61 = 84.8544666810555

x62 = -99.4523023295457

x63 = 93.930178791426

x64 = 28.3057989164392

x65 = -40.1111077617385

x66 = 5.96558449091179

x67 = -57.5644002816818

x68 = 58.3254620507417

x69 = 91.8357836890328

x70 = 341.068800873823

x71 = 79.2694130746736

x72 = 26.211403814046

x73 = 45.7590914363825

x74 = 61.8161205547303

x75 = -68.0363757936478

x76 = 70.1937009643031

x77 = -38.0167126593453

x78 = -77.810219604816

x79 = 63.9105156571235

x80 = -86.1878000143888

x81 = -9.39331292663831

x82 = 76.4768862714827

x83 = -63.8475855888614

x84 = 49.9478816411689

x85 = -31.7335273521657

x86 = -61.7531904864682

x87 = -44.2998979665249

x88 = 10.1543746956982

x89 = 56.2310669483485

x90 = 17.8338234044732

x91 = 24.1170087116528

x92 = 14.3431649004846

x93 = -90.3765902191752

x94 = -70.130770896041

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