log(sin(x))=0 (уравнение)

Найду корень уравнения: log(sin(x))=0

Вы ввели

log(sin(x)) = 0

$$\log{\left(\sin{\left(x \right)} \right)} = 0$$

Подробное решение

Дано уравнение
$$\log{\left(\sin{\left(x \right)} \right)} = 0$$
преобразуем
$$\log{\left(\sin{\left(x \right)} \right)} = 0$$
$$\log{\left(\sin{\left(x \right)} \right)} + 0 = 0$$
Сделаем замену
$$w = \sin{\left(x \right)}$$
Дано уравнение
$$\log{\left(w \right)} = 0$$
$$\log{\left(w \right)} = 0$$
Это уравнение вида:

log(v)=p

По определению log

v=e^p

тогда
$$1 w + 0 = e^{\frac{0}{1}}$$
упрощаем
$$w = 1$$
делаем обратную замену
$$\sin{\left(x \right)} = w$$
Дано уравнение
$$\sin{\left(x \right)} = w$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, где n - любое целое число
подставляем w:

Быстрый ответ

     pi
x1 = --
     2 

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

Сумма и произведение корней

сумма

    pi
0 + --
    2 

$$0 + \frac{\pi}{2}$$

=

pi
--
2 

$$\frac{\pi}{2}$$

произведение

  pi
1*--
  2 

$$1 \frac{\pi}{2}$$

=

pi
--
2 

$$\frac{\pi}{2}$$

Численный ответ

x1 = 20.4203523413041 + 1.47591396943848e-6*i

x2 = 1.57079660167231

x3 = -10.9955735120111

x4 = -29.8451300946193

x5 = -98.9601678624104

x6 = -48.6946869113465

x7 = 58.119464370396

x8 = -73.8274273885517 + 1.5465899417386e-6*i

x9 = 64.4026494793175 + 1.47549472973716e-6*i

x10 = -80.1106127903609 - 1.49593986406504e-6*i

x11 = 89.5353892520407

x12 = 83.2522058199769

x13 = 14.1371668991051 + 1.57912333020909e-6*i

x14 = -48.694686949957 - 3.48332995838883e-7*i

x15 = 51.8362789063255

x16 = -29.8451302446469 + 1.56171152047569e-6*i

x17 = -54.9778713174705 - 4.45451298579927e-7*i

x18 = -23.5619447959101 + 1.47687939679762e-6*i

x19 = -36.1283154154305

x20 = 58.1194640425023 + 1.59909631203725e-6*i

x21 = -54.9778719110131

x22 = 51.8362786093149 - 1.50646716105411e-6*i

x23 = -54.9778705207315 + 5.81575500703856e-7*i

x24 = -17.2787590751963 + 1.47285553940655e-6*i

x25 = -67.5442419322395 + 1.47834743734414e-6*i

x26 = 39.2699074396221

x27 = 76.9690195814988

x28 = -73.827427279653

x29 = -42.4115015246508 - 1.43345550197194e-6*i

x30 = -29.8451302007713 - 1.57918981389732e-6*i

x31 = -10.9955733481573 + 6.5127652579786e-7*i

x32 = 14.1371669434725 - 1.56210516922416e-6*i

x33 = -86.3937986220237 - 1.46376198181665e-6*i

x34 = 32.9867238176799 + 7.23174778366821e-7*i

x35 = 76.9690208439152

x36 = -10.9955741211687 - 2.61532405458509e-7*i

x37 = 26.7035372979479

x38 = -67.5442421737401

x39 = -61.2610561822839 + 1.45762099226605e-6*i

x40 = -42.4115005591739

x41 = 70.6858351446748 + 1.4920648078667e-6*i

x42 = 1.57079557309815 - 1.39597937256977e-6*i

x43 = -61.2610552210018

x44 = 20.4203521458531

x45 = 83.2522046133019

x46 = -86.3937977199047

x47 = 58.1194640878142 - 1.54745309541394e-6*i

x48 = 32.9867230918614 - 6.32762411257027e-8*i

x49 = 32.9867224176774

x50 = -92.676984146129 - 4.09081732918272e-7*i

x51 = -98.9601690759292

x52 = 26.7035380336909 + 1.48414933958316e-6*i

x53 = 14.1371671149845

x54 = -54.9778706878326

x55 = 70.6858344584179

x56 = 76.9690209763474 + 7.96363548114344e-7*i

x57 = -73.8274273446888 - 1.59897178129631e-6*i

x58 = 39.2699074534854 - 2.43024050539207e-7*i

x59 = 89.5353897747913 - 1.29857788236559e-6*i

x60 = -4.71238973521995

x61 = -48.6946856746846

x62 = -36.1283156556098 - 1.49887098218185e-6*i

x63 = -98.9601685005998 - 6.119917163757e-7*i

x64 = 7.85398147767715 - 1.50241514482621e-6*i

x65 = -4.71238851018714

x66 = -80.1106125767506

x67 = -61.2610570233699

x68 = 32.9867236651869

x69 = 64.402649305744

x70 = 95.8185760669056

x71 = 83.2522055855063 + 2.09191766033852e-7*i

x72 = 45.5530926724619 - 1.35103224629959e-6*i

x73 = 95.8185757390985 - 1.51089030280694e-6*i

x74 = 45.5530937626454

x75 = -17.2787598626449

x76 = 7.85398174556756

x77 = -92.6769828388254

x78 = -23.5619450139675

x79 = -4.71238974981597 - 2.92973107103631e-7*i

x80 = 89.5353909237568

x81 = 39.2699086546913

x82 = -92.6769840888992

x83 = 83.2522046592344 - 1.98391556760954e-7*i

x84 = -10.9955747464857

x85 = 45.5530919949765

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