sin(x/3)=-1 (уравнение)

Найду корень уравнения: sin(x/3)=-1

Вы ввели

   /x\     
sin|-| = -1
   \3/     

$$\sin{\left(\frac{x}{3} \right)} = -1$$

Подробное решение

Дано уравнение
$$\sin{\left(\frac{x}{3} \right)} = -1$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$\frac{x}{3} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$\frac{x}{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
Или
$$\frac{x}{3} = 2 \pi n - \frac{\pi}{2}$$
$$\frac{x}{3} = 2 \pi n + \frac{3 \pi}{2}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$\frac{1}{3}$$
получим ответ:
$$x_{1} = 6 \pi n - \frac{3 \pi}{2}$$
$$x_{2} = 6 \pi n + \frac{9 \pi}{2}$$

Сумма и произведение корней

сумма

    3*pi   9*pi
0 - ---- + ----
     2      2  

$$\left(- \frac{3 \pi}{2} + 0\right) + \frac{9 \pi}{2}$$

=

3*pi

$$3 \pi$$

произведение

  -3*pi 9*pi
1*-----*----
    2    2  

$$\frac{9 \pi}{2} \cdot 1 \left(- \frac{3 \pi}{2}\right)$$

=

      2
-27*pi 
-------
   4   

$$- \frac{27 \pi^{2}}{4}$$

Быстрый ответ

     -3*pi
x1 = -----
       2  

$$x_{1} = - \frac{3 \pi}{2}$$

     9*pi
x2 = ----
      2  

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

Численный ответ

x1 = -42.4114988767471

x2 = -23.5619461984625

x3 = -23.5619464082898

x4 = -80.1106140804049

x5 = -61.261055404731

x6 = -4.71238779907246

x7 = -4.71238859426546

x8 = -4.71238743209195

x9 = 14.1371654793596

x10 = 70.6858353577546

x11 = 51.8362797759315

x12 = -4.71239050808881

x13 = 14.1371678181509

x14 = 70.6858329020223

x15 = 32.9867233840103

x16 = 89.5353900126591

x17 = -61.2610582787117

x18 = -61.2610574882809

x19 = 89.5353892972875

x20 = 51.8362773664247

x21 = 14.1371656532869

x22 = 70.6858361981262

x23 = -23.5619482920391

x24 = 89.5353917263916

x25 = 51.8362780425465

x26 = -80.1106134301082

x27 = -4.71238861603651

x28 = -42.4114992857075

x29 = -287.45572872034

x30 = 32.986722548861

x31 = 14.1371684115644

x32 = -61.2610578317089

x33 = -42.4114999779636

x34 = -23.5619458805467

x35 = -98.9601680902976

x36 = -98.9601688529851

x37 = -42.4114997427582

x38 = -4.71239024216915

x39 = -98.9601670486134

x40 = 32.9867213216468

x41 = -61.2610561157384

x42 = -23.56194501223

x43 = -80.110611523969

x44 = 70.6858331755052

x45 = 89.5353914041279

x46 = -80.1106140658652

x47 = -61.2610582551732

x48 = 14.1371660778537

x49 = -80.1106116945809

x50 = 70.6858338911042

x51 = 14.1371665638329

x52 = 51.8362802985196

x53 = -80.1106089100043

x54 = -80.1106111588322

x55 = 14.1371682432467

x56 = -80.1106125777448

x57 = -4.71238845524301

x58 = 32.9867224232349

x59 = -23.5619434901217

x60 = -98.9601689835803

x61 = -42.4115023072805

x62 = 89.5353921684403

x63 = 70.6858360595363

x64 = 89.5353908982368

x65 = -42.4115021671282

x66 = 51.8362789040765

x67 = 70.6858344740862

x68 = 51.8362773818419

x69 = 51.8362823163449

x70 = 14.1371671114619

x71 = -4.71238948689267

x72 = -23.5619434738021

x73 = 32.9867216931265

x74 = -98.960167330773

x75 = 89.5353891331438

x76 = -42.4115005778798

x77 = 89.5353922692829

x78 = -23.5619441460216

x79 = -61.2610570017434

x80 = 32.986722491348

x81 = 70.6858336375702

x82 = 32.9867243978937

x83 = 183.783173900171

x84 = -61.2610552425571

x85 = -23.5619459803956

x86 = 51.8362798839346

x87 = -42.411501460893

x88 = 32.9867241354141

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