2sin(2x-4π)=-√3 (уравнение)

Найду корень уравнения: 2sin(2x-4π)=-√3

Вы ввели

                                        ___
2*sin(2*x - 4*POST_GRBEK_SMALL_pi) = -\/ 3 

$$2 \sin{\left(- 4 POST_{GRBEK SMALL \pi} + 2 x \right)} = - \sqrt{3}$$

Подробное решение

Дано уравнение
$$2 \sin{\left(- 4 POST_{GRBEK SMALL \pi} + 2 x \right)} = - \sqrt{3}$$
- это простейшее тригонометрическое ур-ние

Разделим обе части ур-ния на -2

Ур-ние превратится в
$$\sin{\left(4 POST_{GRBEK SMALL \pi} - 2 x \right)} = \frac{\sqrt{3}}{2}$$
Это ур-ние преобразуется в
$$4 POST_{GRBEK SMALL \pi} - 2 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}$$
$$4 POST_{GRBEK SMALL \pi} - 2 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi$$
Или
$$4 POST_{GRBEK SMALL \pi} - 2 x = 2 \pi n + \frac{\pi}{3}$$
$$4 POST_{GRBEK SMALL \pi} - 2 x = 2 \pi n + \frac{2 \pi}{3}$$
, где n - любое целое число
Перенесём
$$4 POST_{GRBEK SMALL \pi}$$
в правую часть ур-ния
с противоположным знаком, итого:
$$- 2 x = - 4 POST_{GRBEK SMALL \pi} + 2 \pi n + \frac{\pi}{3}$$
$$- 2 x = - 4 POST_{GRBEK SMALL \pi} + 2 \pi n + \frac{2 \pi}{3}$$
Разделим обе части полученного ур-ния на
$$-2$$
получим ответ:
$$x_{1} = 2 POST_{GRBEK SMALL \pi} - \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 POST_{GRBEK SMALL \pi} - \pi n - \frac{\pi}{3}$$

Сумма и произведение корней

сумма

                            pi                           pi
0 + 2*POST_GRBEK_SMALL_pi - -- + 2*POST_GRBEK_SMALL_pi - --
                            3                            6 

$$\left(2 POST_{GRBEK SMALL \pi} - \frac{\pi}{6}\right) + \left(\left(2 POST_{GRBEK SMALL \pi} - \frac{\pi}{3}\right) + 0\right)$$

=

                        pi
4*POST_GRBEK_SMALL_pi - --
                        2 

$$4 POST_{GRBEK SMALL \pi} - \frac{\pi}{2}$$

произведение

  /                        pi\ /                        pi\
1*|2*POST_GRBEK_SMALL_pi - --|*|2*POST_GRBEK_SMALL_pi - --|
  \                        3 / \                        6 /

$$1 \cdot \left(2 POST_{GRBEK SMALL \pi} - \frac{\pi}{3}\right) \left(2 POST_{GRBEK SMALL \pi} - \frac{\pi}{6}\right)$$

=

(-pi + 6*POST_GRBEK_SMALL_pi)*(-pi + 12*POST_GRBEK_SMALL_pi)
------------------------------------------------------------
                             18                             

$$\frac{\left(6 POST_{GRBEK SMALL \pi} - \pi\right) \left(12 POST_{GRBEK SMALL \pi} - \pi\right)}{18}$$

Быстрый ответ

                             pi
x1 = 2*POST_GRBEK_SMALL_pi - --
                             3 

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

                             pi
x2 = 2*POST_GRBEK_SMALL_pi - --
                             6 

$$x_{2} = 2 POST_{GRBEK SMALL \pi} - \frac{\pi}{6}$$

Численный ответ

x1 = -45.0294947014537

x2 = 40.317105721069

x3 = -1.0471975511966

x4 = -22.5147473507269

x5 = 5.75958653158129

x6 = -51.3126800086333

x7 = -72.7802298081635

x8 = 15.1843644923507

x9 = -3.66519142918809

x10 = -29.3215314335047

x11 = 90.0589894029074

x12 = 36.6519142918809

x13 = 43.4586983746588

x14 = -7.33038285837618

x15 = 80.634211442138

x16 = 84.2994028713261

x17 = 42.9350995990605

x18 = -53.9306738866248

x19 = 58.6430628670095

x20 = 37.1755130674792

x21 = -261.799387799149

x22 = -4.18879020478639

x23 = 8.37758040957278

x24 = 30.3687289847013

x25 = 39900.3210956928

x26 = -31.9395253114962

x27 = -0.523598775598299

x28 = 62.3082542961976

x29 = -60.2138591938044

x30 = 18.3259571459405

x31 = -73.3038285837618

x32 = 27.7507351067098

x33 = -16.2315620435473

x34 = -16.7551608191456

x35 = 65.4498469497874

x36 = 68.0678408277789

x37 = -94.7713783832921

x38 = 64.9262481741891

x39 = -57.5958653158129

x40 = 71.2094334813686

x41 = -69.6386371545737

x42 = -79.5870138909414

x43 = -60.7374579694027

x44 = -67.0206432765823

x45 = -242.426233102012

x46 = -19.8967534727354

x47 = 34.0339204138894

x48 = 14.6607657167524

x49 = -85.870199198121

x50 = 74.3510261349584

x51 = -66.497044500984

x52 = 78.0162175641465

x53 = -23.0383461263252

x54 = -91.6297857297023

x55 = 8.90117918517108

x56 = 52.3598775598299

x57 = -25.6563400043166

x58 = 86.9173967493176

x59 = 46.0766922526503

x60 = -88.4881930761125

x61 = -97.9129710368819

x62 = -82.2050077689329

x63 = -44.5058959258554

x64 = 49.7418836818384

x65 = 100.007366139275

x66 = -89.0117918517108

x67 = -47.6474885794452

x68 = 68.5914396033772

x69 = 124.616508592395

x70 = -13.6135681655558

x71 = 56.025068989018

x72 = 96.342174710087

x73 = -28.7979326579064

x74 = -6.80678408277789

x75 = -50.789081233035

x76 = 21.4675497995303

x77 = 71.733032256967

x78 = -1257.16066021152

x79 = 12.0427718387609

x80 = -63.8790506229925

x81 = -38.2227106186758

x82 = 24.0855436775217

x83 = -95.2949771588904

x84 = -35.6047167406843

x85 = 87.4409955249159

x86 = 93.7241808320955

x87 = -75.9218224617533

x88 = 20.943951023932

x89 = -41.8879020478639

x90 = 2.0943951023932

x91 = -9.94837673636768

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