cos(3x)=(1/2) (уравнение)

Найду корень уравнения: cos(3x)=(1/2)

Вы ввели

cos(3*x) = 1/2

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

Подробное решение

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

Быстрый ответ

     pi
x1 = --
     9 

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

     5*pi
x2 = ----
      9  

$$x_{2} = \frac{5 \pi}{9}$$

Сумма и произведение корней

сумма

    pi   5*pi
0 + -- + ----
    9     9  

$$\left(0 + \frac{\pi}{9}\right) + \frac{5 \pi}{9}$$

=

2*pi
----
 3  

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

произведение

  pi 5*pi
1*--*----
  9   9  

$$\frac{5 \pi}{9} \cdot 1 \frac{\pi}{9}$$

=

    2
5*pi 
-----
  81 

$$\frac{5 \pi^{2}}{81}$$

Численный ответ

x1 = 31.7649923862968

x2 = 12.2173047639603

x3 = -75.0491578357562

x4 = 82.0304748437335

x5 = -71.5584993317675

x6 = 44.331363000656

x7 = 64.5771823237902

x8 = 62.482787221397

x9 = -31.7649923862968

x10 = 58.2939970166106

x11 = 10.1229096615671

x12 = -56.1996019142174

x13 = 98.0875039620813

x14 = -33.85938748869

x15 = -100.181899064475

x16 = 56.1996019142174

x17 = 40.1425727958696

x18 = -23.387411976724

x19 = 47.8220215046446

x20 = -67.3697091269811

x21 = 54.1052068118242

x22 = -3.83972435438753

x23 = -75.7472895365539

x24 = -27.5762021815104

x25 = -43.6332312998582

x26 = 38.0481776934764

x27 = -52.010811709431

x28 = -77.8416846389471

x29 = 16.4060949687467

x30 = 73.6528944341607

x31 = 26.8780704807127

x32 = -25.4818070791172

x33 = 8.02851455917392

x34 = -98.0875039620813

x35 = -82.0304748437335

x36 = -58.2939970166106

x37 = 93.8987137572949

x38 = -104.370689269261

x39 = -93.8987137572949

x40 = -14.3116998663535

x41 = 52.010811709431

x42 = 88.3136601509131

x43 = 3.83972435438753

x44 = 49.9164166070378

x45 = 84.1248699461267

x46 = 90.4080552533063

x47 = -2150248.78004967

x48 = -73.6528944341607

x49 = -29.6705972839036

x50 = -95.9931088596881

x51 = -79.9360797413403

x52 = -60.3883921190038

x53 = 35.9537825910832

x54 = 77.8416846389471

x55 = 14.3116998663535

x56 = -1.74532925199433

x57 = -16.4060949687467

x58 = -47.8220215046446

x59 = -8.02851455917392

x60 = -91.8043186549017

x61 = -12.915436464758

x62 = -40.1425727958696

x63 = -54.1052068118242

x64 = 23.387411976724

x65 = 20.5948851735331

x66 = -84.1248699461267

x67 = -49.9164166070378

x68 = -38.0481776934764

x69 = 29.6705972839036

x70 = 5.93411945678072

x71 = -45.7276264022514

x72 = -5.93411945678072

x73 = 18.5004900711399

x74 = -69.4641042293743

x75 = -35.9537825910832

x76 = 95.9931088596881

x77 = 33.85938748869

x78 = 75.7472895365539

x79 = -20.5948851735331

x80 = -89.7099235525085

x81 = 79.9360797413403

x82 = 100.181899064475

x83 = -10.1229096615671

x84 = 0.349065850398866

x85 = 46.4257581030492

x86 = 86.2192650485199

x87 = -12.2173047639603

x88 = -87.6155284501153

x89 = 60.3883921190038

x90 = 42.2369678982628

x91 = -68.7659725285766

x92 = 70.8603676309698

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