1+sin(π-x)=0 (уравнение)

Найду корень уравнения: 1+sin(π-x)=0

Вы ввели

1 + sin(POST_GRBEK_SMALL_pi - x) = 0

$$\sin{\left(POST_{GRBEK SMALL \pi} - x \right)} + 1 = 0$$

Подробное решение

Дано уравнение
$$\sin{\left(POST_{GRBEK SMALL \pi} - x \right)} + 1 = 0$$
- это простейшее тригонометрическое ур-ние

Перенесём 1 в правую часть ур-ния

с изменением знака при 1

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

Быстрый ответ

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

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

                           pi
x2 = POST_GRBEK_SMALL_pi + --
                           2 

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

Сумма и произведение корней

сумма

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

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

=

-pi + 2*POST_GRBEK_SMALL_pi

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

произведение

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

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

=

(pi + 2*POST_GRBEK_SMALL_pi)*(-3*pi + 2*POST_GRBEK_SMALL_pi)
------------------------------------------------------------
                             4                              

$$\frac{\left(2 POST_{GRBEK SMALL \pi} - 3 \pi\right) \left(2 POST_{GRBEK SMALL \pi} + \pi\right)}{4}$$

Численный ответ

x1 = 23.5619444059921

x2 = 80.1106130902139

x3 = 48.6946859012172

x4 = -76.9690195738024

x5 = 36.1283150875497

x6 = -58.1194639976905

x7 = 4.71238874329685

x8 = -64.4026498988255

x9 = -58.1194639046052

x10 = -39.2699084145515

x11 = -64.4026502975618

x12 = 73.8274268520838

x13 = -89.535390750197

x14 = -45.5530929624673

x15 = -20.4203520060805

x16 = 86.3937978869933

x17 = 67.5442408278864

x18 = -26.7035372004893

x19 = -70.6858343571487

x20 = -39.2699076683741

x21 = -1.57079581340397

x22 = -76.9690203748894

x23 = 67.54424230971

x24 = -51.8362791922783

x25 = 42.4115013353669

x26 = -14.1371674455661

x27 = 80.1106131368654

x28 = 86.3937978309099

x29 = 98.9601682515978

x30 = -83.2522055723275

x31 = 54.9778718908148

x32 = -20.4203527465087

x33 = 80.1106122287081

x34 = -83.2522048211133

x35 = 4.71239022926564

x36 = -45.5530935911043

x37 = 48.6946866365921

x38 = -89.5353906059052

x39 = 86.3937984838325

x40 = -7.85398119154045

x41 = 92.6769843439965

x42 = 538.783139388541

x43 = 73.8274274830848

x44 = 17.2787599560783

x45 = -70.6858351534454

x46 = 29.8451297031011

x47 = -14.1371667858125

x48 = 98.9601690454399

x49 = -70.6858331259916

x50 = -7.85398149665124

x51 = 36.1283157235346

x52 = -32.9867224188086

x53 = 48.6946870830469

x54 = -20.420353265929

x55 = 29.8451303231501

x56 = 23.5619437708833

x57 = 42.4115007162407

x58 = -95.8185758680502

x59 = -1.57079639503667

x60 = -83.2522042893833

x61 = 42.4115007274741

x62 = -7.85398205280014

x63 = -1.57079643188553

x64 = -89.5353901118113

x65 = 36.1283159497235

x66 = -14.1371668370864

x67 = 61.2610563112167

x68 = -51.8362783335234

x69 = -64.4026491641039

x70 = 98.9601692809083

x71 = 61.2610571125526

x72 = 10.9955747360645

x73 = 23.5619451518571

x74 = -95.8185763308148

x75 = 29.845130330036

x76 = 10.9955739381756

x77 = -58.1194645939029

x78 = -26.7035379986821

x79 = -45.5530935025548

x80 = 92.6769837888103

x81 = -102.101761026058

x82 = 17.2787591562062

x83 = -32.9867232184024

x84 = -39.2699069219675

x85 = 54.9778710948428

x86 = 4.7123894841958

x87 = 48.6946873020308

x88 = 92.6769830592094

x89 = 73.8274274426229

x90 = -51.8362786893284

x91 = 67.5442415586719

x92 = -95.818575476176

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