cos(2*x+pi/4)=1 (уравнение)

Найду корень уравнения: cos(2*x+pi/4)=1

Вы ввели

   /      pi\    
cos|2*x + --| = 1
   \      4 /    

$$\cos{\left(2 x + \frac{\pi}{4} \right)} = 1$$

Подробное решение

Дано уравнение
$$\cos{\left(2 x + \frac{\pi}{4} \right)} = 1$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$2 x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$2 x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Или
$$2 x + \frac{\pi}{4} = \pi n$$
$$2 x + \frac{\pi}{4} = \pi n - \pi$$
, где n - любое целое число
Перенесём
$$\frac{\pi}{4}$$
в правую часть ур-ния
с противоположным знаком, итого:
$$2 x = \pi n - \frac{\pi}{4}$$
$$2 x = \pi n - \frac{5 \pi}{4}$$
Разделим обе части полученного ур-ния на
$$2$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} - \frac{\pi}{8}$$
$$x_{2} = \frac{\pi n}{2} - \frac{5 \pi}{8}$$

Быстрый ответ

     -pi 
x1 = ----
      8  

$$x_{1} = - \frac{\pi}{8}$$

     7*pi
x2 = ----
      8  

$$x_{2} = \frac{7 \pi}{8}$$

Сумма и произведение корней

сумма

    pi   7*pi
0 - -- + ----
    8     8  

$$\left(- \frac{\pi}{8} + 0\right) + \frac{7 \pi}{8}$$

=

3*pi
----
 4  

$$\frac{3 \pi}{4}$$

произведение

  -pi  7*pi
1*----*----
   8    8  

$$\frac{7 \pi}{8} \cdot 1 \left(- \frac{\pi}{8}\right)$$

=

     2
-7*pi 
------
  64  

$$- \frac{7 \pi^{2}}{64}$$

Численный ответ

x1 = -60.0829594298968

x2 = 93.8550806320245

x3 = 37.3064130181831

x4 = 9.03207901629355

x5 = -38.091810849147

x6 = 87.5718954371561

x7 = -97.7820713167551

x8 = -50.6581813631871

x9 = 5.89048631181155

x10 = 49.8727834721674

x11 = -78.9325156574046

x12 = 31.0232272090657

x13 = -12.9590695481742

x14 = 43.589598281813

x15 = 59.2975613523085

x16 = 90.7134876601913

x17 = -75.7909227332505

x18 = 96.9966729313922

x19 = -78.9325152464504

x20 = -63.2245523814864

x21 = 68.7223390826016

x22 = 84.4303024618006

x23 = -50.6581812825063

x24 = 34.1648202002422

x25 = 15.3152642571177

x26 = 40.448005303171

x27 = 75.0055243571033

x28 = -91.498886171464

x29 = 37.3064128034015

x30 = 24.7400419276239

x31 = 18.4568567239364

x32 = 96.9966732636524

x33 = 12.1736716102657

x34 = -56.941366679619

x35 = -22.3838474516909

x36 = 27.8816348920574

x37 = 78.1471173903386

x38 = 31.0232275798129

x39 = -12.9590699381886

x40 = -34.9502181135041

x41 = -28.667032826468

x42 = 71.8639320521531

x43 = 100.138265993104

x44 = -31.8086255638994

x45 = -72.6493298579695

x46 = -66.3661446076792

x47 = -60.0829590749567

x48 = 53.0143757829915

x49 = -85.2157009585792

x50 = -0.392698873791919

x51 = -94.6404784335422

x52 = -34.9502185114952

x53 = -100.923663813937

x54 = -41.2334038043182

x55 = -3.53429185524601

x56 = -82.0741080108586

x57 = 46.7311908066772

x58 = -6.67588413192134

x59 = -19.2422546233015

x60 = 56.1559687931787

x61 = -56.9413670845717

x62 = -88.3572932833187

x63 = -69.5077375924855

x64 = 81.2887099035841

x65 = 9.03207863533546

x66 = -31.8086252953578

x67 = -28.6670327071557

x68 = 81.288710167773

x69 = 65.5807468595177

x70 = -44.3749960296535

x71 = -94.640478425127

x72 = -47.5165890134565

x73 = 75.0055247036075

x74 = 62.4391538824588

x75 = -16.1006622685843

x76 = 68.7223395780101

x77 = -88.3572931857676

x78 = 53.014376142287

x79 = 2.74889335023851

x80 = 46.7311905050787

x81 = -25.5254404343768

x82 = 24.7400422413392

x83 = 21.5984497040429

x84 = 15.3152644431929

x85 = -53.7997741490291

x86 = -72.6493298961485

x87 = -6.67588428626859

x88 = -19.242255227076

x89 = -9.8174769775948

x90 = 59.2975615930447

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