cos5x=1/2 (уравнение)

Найду корень уравнения: cos5x=1/2

Вы ввели

cos(5*x) = 1/2

$$\cos{\left(5 x \right)} = \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(5 x \right)} = \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$5 x = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$5 x = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Или
$$5 x = \pi n + \frac{\pi}{3}$$
$$5 x = \pi n - \frac{2 \pi}{3}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$5$$
получим ответ:
$$x_{1} = \frac{\pi n}{5} + \frac{\pi}{15}$$
$$x_{2} = \frac{\pi n}{5} - \frac{2 \pi}{15}$$

Быстрый ответ

     pi
x1 = --
     15

$$x_{1} = \frac{\pi}{15}$$

     pi
x2 = --
     3 

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

Сумма и произведение корней

сумма

    pi   pi
0 + -- + --
    15   3 

$$\left(0 + \frac{\pi}{15}\right) + \frac{\pi}{3}$$

=

2*pi
----
 5  

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

произведение

  pi pi
1*--*--
  15 3 

$$\frac{\pi}{3} \cdot 1 \frac{\pi}{15}$$

=

  2
pi 
---
 45

$$\frac{\pi^{2}}{45}$$

Численный ответ

x1 = 22.4100275956072

x2 = -73.9321471144798

x3 = 70.162235930172

x4 = 64.2979296434711

x5 = 98.2271303022409

x6 = 94.0383400974545

x7 = -3.97935069454707

x8 = -81.8908485035739

x9 = 71.8377520120866

x10 = 56.339228254377

x11 = 58.0147443362915

x12 = 0.20943951023932

x13 = -7.74926187885482

x14 = 34.1386401690091

x15 = 12.3569311041199

x16 = 78.1209373192662

x17 = 90.2684289131467

x18 = -63.8790506229925

x19 = 2.30383461263252

x20 = -17.8023583703422

x21 = 66.3923247458643

x22 = 50.0560429471974

x23 = -51.7315590291119

x24 = -37.4896723328382

x25 = 100.321525404634

x26 = 16.1268422884276

x27 = 54.2448331519838

x28 = -50.0560429471974

x29 = 37.9085513533168

x30 = 80.2153324216594

x31 = -68.0678408277789

x32 = -37.9085513533168

x33 = -53.8259541315051

x34 = -12.3569311041199

x35 = 29.9498499642227

x36 = 6.07374579694027

x37 = 24.0855436775217

x38 = 36.2330352714023

x39 = 40.00294645571

x40 = -12.7758101245985

x41 = 17.8023583703422

x42 = -94.0383400974545

x43 = 68.0678408277789

x44 = 27.8554548618295

x45 = -85.6607596878817

x46 = -60.1091394386847

x47 = -91.9439449950613

x48 = 81.8908485035739

x49 = -43.7728576400178

x50 = 73.9321471144798

x51 = -70.162235930172

x52 = -47.9616478448042

x53 = -65.554566704907

x54 = -77.7020582987875

x55 = -58.0147443362915

x56 = -61.7846555205993

x57 = -95.713856179369

x58 = -220.120925261525

x59 = -40.00294645571

x60 = 44.1917366604964

x61 = -97.8082512817622

x62 = -29.9498499642227

x63 = -80.634211442138

x64 = 76.4454212373516

x65 = 61.3657765001206

x66 = 46.2861317628896

x67 = -87.7551547902749

x68 = -21.5722695546499

x69 = 60.1091394386847

x70 = 47.9616478448042

x71 = 91.9439449950613

x72 = -27.8554548618295

x73 = -26.1799387799149

x74 = -6.07374579694027

x75 = 83.9852436059671

x76 = 88.1740338107535

x77 = 32.8820031075732

x78 = -83.9852436059671

x79 = -98.2271303022409

x80 = -75.6076631963944

x81 = 3.97935069454707

x82 = -31.6253660461373

x83 = -16.1268422884276

x84 = -71.8377520120866

x85 = 10.2625360017267

x86 = -19.8967534727354

x87 = 20.315632493214

x88 = -41.6784625376246

x89 = -9.84365698124802

x90 = 26.1799387799149

x91 = -33.7197611485304

x92 = 14.0324471860344

x93 = -14.0324471860344

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