sin5x=-1 (уравнение)

Найду корень уравнения: sin5x=-1

Вы ввели

sin(5*x) = -1

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

Подробное решение

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

Быстрый ответ

     -pi 
x1 = ----
      10 

$$x_{1} = - \frac{\pi}{10}$$

     3*pi
x2 = ----
      10 

$$x_{2} = \frac{3 \pi}{10}$$

Сумма и произведение корней

сумма

    pi   3*pi
0 - -- + ----
    10    10 

$$\left(- \frac{\pi}{10} + 0\right) + \frac{3 \pi}{10}$$

=

pi
--
5 

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

произведение

  -pi  3*pi
1*----*----
   10   10 

$$\frac{3 \pi}{10} \cdot 1 \left(- \frac{\pi}{10}\right)$$

=

     2
-3*pi 
------
 100  

$$- \frac{3 \pi^{2}}{100}$$

Численный ответ

x1 = -15.3938039659798

x2 = 70.0575160400951

x3 = 46.1814120735606

x4 = -21.6769893550399

x5 = 18.5353965790101

x6 = -54.3495529216771

x7 = 39.8982267733088

x8 = 5.96902610606135

x9 = 42.411500742904

x10 = 53.721234458049

x11 = 9.73893731490784

x12 = -71.942471902222

x13 = 83.8805239304316

x14 = -44.2964563449265

x15 = 97.703531597323

x16 = -278.030949800724

x17 = 90.163709249367

x18 = 66.2876049654061

x19 = -14.1371668558998

x20 = 8.48230007440474

x21 = -69.42919768965

x22 = 76.3407014282439

x23 = -92.0486647063443

x24 = -45.5530935651178

x25 = 62.5176937218139

x26 = -24.1902633373992

x27 = -78.2256569789261

x28 = 19.7920338136899

x29 = -48.0663675448867

x30 = -85.7654794309647

x31 = -64.4026495871078

x32 = -79.482294220441

x33 = 96.4468943771426

x34 = -99.588487199358

x35 = 60.0044196780723

x36 = 26.0752188396735

x37 = 52.4645972249968

x38 = -41.7831822653761

x39 = -11.6238928872839

x40 = -1.57079641167075

x41 = -89.5353907172626

x42 = -38.0132710938785

x43 = -7.85398151412802

x44 = -55.6061900437796

x45 = 72.570790252592

x46 = -59.376101032934

x47 = 100.216805676894

x48 = -98.3318500297616

x49 = 16.0221225012125

x50 = 77.5973383555507

x51 = 82.6238869092536

x52 = 63.7743309633316

x53 = 86.393797898483

x54 = -51.8362786929109

x55 = -58.1194640118314

x56 = -22.9336264075169

x57 = 73.827427452472

x58 = -88.2787534842036

x59 = 5.96902607240456

x60 = 33.6150412025629

x61 = 29.8451302988705

x62 = -81.9955682642019

x63 = 32.3584042700694

x64 = 12.2522113473678

x65 = 28.5884931335501

x66 = 80.1106129684713

x67 = -35.4999970742626

x68 = 2.19911490246744

x69 = -4.08407038508313

x70 = 93.933620397379

x71 = -31.7300858152792

x72 = -61.8893749736422

x73 = 49.9513232359226

x74 = 43.6681379124747

x75 = 38.6415898747305

x76 = -25.4469005589667

x77 = -0.314159211507129

x78 = -4.0840703117445

x79 = -68.1725604897886

x80 = 22.3053078047562

x81 = -34.2433598309698

x82 = 87.6504350205451

x83 = -10.3672558405688

x84 = -65.6592865140204

x85 = -75.7123829768599

x86 = -27.9601746876698

x87 = 56.2345085113038

x88 = -95.8185758687136

График