cos(x)=1/√5 (уравнение)

Найду корень уравнения: cos(x)=1/√5

Вы ввели

             1  
cos(x) = 1*-----
             ___
           \/ 5 

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

Подробное решение

Дано уравнение
$$\cos{\left(x \right)} = 1 \cdot \frac{1}{\sqrt{5}}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = \pi n + \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)}$$
Или
$$x = \pi n + \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)}$$
$$x = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)}$$
, где n - любое целое число

Быстрый ответ

           /  ___\       
           |\/ 5 |       
x1 = - acos|-----| + 2*pi
           \  5  /       

$$x_{1} = - \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)} + 2 \pi$$

         /  ___\
         |\/ 5 |
x2 = acos|-----|
         \  5  /

$$x_{2} = \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)}$$

Сумма и произведение корней

сумма

          /  ___\              /  ___\
          |\/ 5 |              |\/ 5 |
0 + - acos|-----| + 2*pi + acos|-----|
          \  5  /              \  5  /

$$\operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)} + \left(0 + \left(- \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)} + 2 \pi\right)\right)$$

=

2*pi

$$2 \pi$$

произведение

  /      /  ___\       \     /  ___\
  |      |\/ 5 |       |     |\/ 5 |
1*|- acos|-----| + 2*pi|*acos|-----|
  \      \  5  /       /     \  5  /

$$1 \left(- \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)}$$

=

/      /  ___\       \     /  ___\
|      |\/ 5 |       |     |\/ 5 |
|- acos|-----| + 2*pi|*acos|-----|
\      \  5  /       /     \  5  /

$$\left(- \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{\sqrt{5}}{5} \right)}$$

Численный ответ

x1 = -49.1583337396426

x2 = -36.5919631252834

x3 = 76.5053724039491

x4 = -68.0078896611814

x5 = -93.1406308898997

x6 = 5.1760365893855

x7 = 45.0894458680512

x8 = 61.7247043540018

x9 = -17.7424072037447

x10 = -42.875148432463

x11 = 55.4415190468222

x12 = -82.7885577111287

x13 = 63.93900178959

x14 = 57.6558164824104

x15 = 49.1583337396426

x16 = -86.8574455827201

x17 = -63.93900178959

x18 = 42.875148432463

x19 = -32.523075253692

x20 = 93.1406308898997

x21 = -26.2398899465124

x22 = 74.2910749683609

x23 = 32.523075253692

x24 = 80.5742602755405

x25 = -55.4415190468222

x26 = 26.2398899465124

x27 = 13.6735193321533

x28 = -19.9567046393328

x29 = 30.3087778181038

x30 = 95.3549283254879

x31 = -24.0255925109243

x32 = 68.0078896611814

x33 = -95.3549283254879

x34 = -30.3087778181038

x35 = 82.7885577111287

x36 = -99.4238161970793

x37 = 89.0717430183083

x38 = -74.2910749683609

x39 = -45.0894458680512

x40 = -11.4592218965651

x41 = 99.4238161970793

x42 = 11.4592218965651

x43 = -7.39033402497368

x44 = 19.9567046393328

x45 = 36.5919631252834

x46 = 24.0255925109243

x47 = 38.8062605608716

x48 = 17.7424072037447

x49 = 51.3726311752308

x50 = 7.39033402497368

x51 = 101.638113632667

x52 = 70.2221870967695

x53 = -57.6558164824104

x54 = 1.10714871779409

x55 = -5.1760365893855

x56 = -80.5742602755405

x57 = -70.2221870967695

x58 = -61.7247043540018

x59 = 86.8574455827201

x60 = -76.5053724039491

x61 = -1.10714871779409

x62 = -89.0717430183083

x63 = -38.8062605608716

x64 = -13.6735193321533

x65 = -51.3726311752308

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