sin(x/5)=-1 (уравнение)

Найду корень уравнения: sin(x/5)=-1

Вы ввели

   /x\     
sin|-| = -1
   \5/     

$$\sin{\left(\frac{x}{5} \right)} = -1$$

Подробное решение

Дано уравнение
$$\sin{\left(\frac{x}{5} \right)} = -1$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$\frac{x}{5} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$\frac{x}{5} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Или
$$\frac{x}{5} = 2 \pi n - \frac{\pi}{2}$$
$$\frac{x}{5} = 2 \pi n + \frac{3 \pi}{2}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$\frac{1}{5}$$
получим ответ:
$$x_{1} = 10 \pi n - \frac{5 \pi}{2}$$
$$x_{2} = 10 \pi n + \frac{15 \pi}{2}$$

Быстрый ответ

     -5*pi
x1 = -----
       2  

$$x_{1} = - \frac{5 \pi}{2}$$

     15*pi
x2 = -----
       2  

$$x_{2} = \frac{15 \pi}{2}$$

Сумма и произведение корней

сумма

    5*pi   15*pi
0 - ---- + -----
     2       2  

$$\left(- \frac{5 \pi}{2} + 0\right) + \frac{15 \pi}{2}$$

=

5*pi

$$5 \pi$$

произведение

  -5*pi 15*pi
1*-----*-----
    2     2  

$$\frac{15 \pi}{2} \cdot 1 \left(- \frac{5 \pi}{2}\right)$$

=

      2
-75*pi 
-------
   4   

$$- \frac{75 \pi^{2}}{4}$$

Численный ответ

x1 = 54.9778739070868

x2 = -7.85398252907437

x3 = -70.685837162213

x4 = 54.9778689186669

x5 = -70.6858327153891

x6 = 86.3938002605166

x7 = 23.5619451608833

x8 = -70.6858340990446

x9 = 54.9778710685943

x10 = 86.3938005445281

x11 = -70.6858356915798

x12 = -39.26990668074

x13 = -39.2699038743622

x14 = -70.685836090489

x15 = 54.9778739589187

x16 = -102.10175887468

x17 = 23.5619408834841

x18 = 54.9778673272457

x19 = 23.5619473347874

x20 = -70.6858321762027

x21 = 23.5619443808614

x22 = -39.2699056225628

x23 = -7.85398044420763

x24 = 54.9778699787461

x25 = 86.3937959795513

x26 = -70.6858403532579

x27 = -39.2699107183389

x28 = 86.3938002259124

x29 = -39.2699060064999

x30 = -70.6858372210573

x31 = -70.6858352294967

x32 = -39.2699075615161

x33 = 54.9778689688703

x34 = 54.9778701894147

x35 = 86.3937969712361

x36 = -39.2699098919478

x37 = -70.685834328039

x38 = 54.9778728294449

x39 = 23.5619468123787

x40 = 54.9778773290813

x41 = -39.2699100741378

x42 = 54.9778694535016

x43 = 23.5619423606603

x44 = 86.3937978865907

x45 = 86.3938002774446

x46 = 23.5619474594851

x47 = 23.5619427443373

x48 = 54.9778724916688

x49 = 86.3937987671659

x50 = -7.85398333580381

x51 = -70.6858331974729

x52 = -39.2699084230805

x53 = 54.9778735305754

x54 = -7.85398243950208

x55 = -70.6858334503184

x56 = -39.269907526638

x57 = -102.101763335628

x58 = 23.5619460544654

x59 = -7.85398378062012

x60 = 23.561944265976

x61 = 86.3937961837718

x62 = 86.3937977643927

x63 = -7.85398073158663

x64 = -39.2699093152787

x65 = 86.3937956122847

x66 = -7.85397932187207

x67 = -7.8539799256472

x68 = 86.3937954049156

x69 = 86.3937995909895

x70 = -102.101760231949

x71 = 23.561943419582

x72 = -39.2699057643872

x73 = -39.2699105971761

x74 = -102.10175866509

x75 = -7.85397946359087

x76 = -70.6858343297935

x77 = -7.85398394048364

x78 = -7.85398149604924

x79 = -7.85398266198407

x80 = -70.6858367926279

x81 = 54.9778719687294

x82 = 23.5619425116314

x83 = 23.5619466631903

x84 = -7.85398237386435

x85 = -70.6858322311967

x86 = -7.85398420522347

x87 = -7.85397905565661

x88 = -102.101759445353

x89 = -102.101759218319

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