sin(4x)=-1/2 (уравнение)

Найду корень уравнения: sin(4x)=-1/2

Вы ввели

sin(4*x) = -1/2

$$\sin{\left(4 x \right)} = - \frac{1}{2}$$

Подробное решение

Дано уравнение
$$\sin{\left(4 x \right)} = - \frac{1}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$4 x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Или
$$4 x = 2 \pi n - \frac{\pi}{6}$$
$$4 x = 2 \pi n + \frac{7 \pi}{6}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{2} - \frac{\pi}{24}$$
$$x_{2} = \frac{\pi n}{2} + \frac{7 \pi}{24}$$

Быстрый ответ

     -pi 
x1 = ----
      24 

$$x_{1} = - \frac{\pi}{24}$$

     7*pi
x2 = ----
      24 

$$x_{2} = \frac{7 \pi}{24}$$

Сумма и произведение корней

сумма

    pi   7*pi
0 - -- + ----
    24    24 

$$\left(- \frac{\pi}{24} + 0\right) + \frac{7 \pi}{24}$$

=

pi
--
4 

$$\frac{\pi}{4}$$

произведение

  -pi  7*pi
1*----*----
   24   24 

$$\frac{7 \pi}{24} \cdot 1 \left(- \frac{\pi}{24}\right)$$

=

     2
-7*pi 
------
 576  

$$- \frac{7 \pi^{2}}{576}$$

Численный ответ

x1 = -83.9067037896274

x2 = 84.1685031774265

x3 = 98.3056701185806

x4 = 10.3410758180664

x5 = -91.7606854236019

x6 = -47.7783882733448

x7 = -85.4775001164223

x8 = 7.72308194007491

x9 = 54.3233729683235

x10 = -88.0954939944138

x11 = -1.70169602069447

x12 = -37.8300115369771

x13 = 62.177354602298

x14 = 57.9885643975116

x15 = 51.1817803147337

x16 = 40.1862060271694

x17 = 64.2717497046912

x18 = 50.1345827635371

x19 = -17.9332580642417

x20 = 94.1168799137942

x21 = -59.8211601121056

x22 = -10.0792764302673

x23 = 21.860248881229

x24 = -117.940624203517

x25 = -0.130899693899575

x26 = -81.8123086872342

x27 = 46.469391334349

x28 = 76.8381203190504

x29 = 18.1950574520409

x30 = -54.0615735805244

x31 = -89.6662903212087

x32 = 6.15228561328001

x33 = -39.9244066393703

x34 = -22.1220482690281

x35 = 2.48709418409192

x36 = -99.6146670575764

x37 = 90.4516884846061

x38 = 1.43989663289532

x39 = -45.6839931709516

x40 = 43.8513974563575

x41 = 19.7658537788358

x42 = 11.9118721448613

x43 = 32.332224393195

x44 = -69.7695368484733

x45 = 70.0313362362725

x46 = -15.8388629618485

x47 = 79.9797129726402

x48 = -3.79609112308767

x49 = -33.6412213321907

x50 = 65.8425460314861

x51 = 68.4605399094776

x52 = -73.9583270532597

x53 = -14.2680666350536

x54 = -61.9155552144988

x55 = 41.7570023539643

x56 = 24.4782427592205

x57 = 73.1729288898623

x58 = -80.2415123604393

x59 = 26.0490390860154

x60 = -63.4863515412937

x61 = -32.0704250053958

x62 = -76.0527221556529

x63 = 72.1257313386657

x64 = -29.9760299030026

x65 = -7.98488132787406

x66 = 86.2628982798197

x67 = -19.5040543910366

x68 = 92.022484811401

x69 = 2838.29806282448

x70 = 76.3145215434521

x71 = -98.0438707307815

x72 = -44.1131968441567

x73 = -25.7872396982162

x74 = -41.4952029661652

x75 = 95.1640774649908

x76 = 42.2806011295626

x77 = -58.2503637853107

x78 = 4.05789051088682

x79 = 87.8336946066146

x80 = -55.6323699073193

x81 = -36.2592152101822

x82 = 28.1434341884086

x83 = 14.0062672472545

x84 = -23.692844595823

x85 = 33.9030207199899

x86 = -51.9671784781312

x87 = -67.6751417460801

x88 = 48.0401876611439

x89 = 55.8941692951184

x90 = -11.6500727570621

x91 = -66.1043454192852

x92 = -77.6235184824478

x93 = 20.2894525544341

x94 = -95.9494756283883

x95 = 35.997415822383

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