sin(x)=3/pi (уравнение)

Найду корень уравнения: sin(x)=3/pi

Вы ввели

         3 
sin(x) = --
         pi

$$\sin{\left(x \right)} = \frac{3}{\pi}$$

Подробное решение

Дано уравнение
$$\sin{\left(x \right)} = \frac{3}{\pi}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{3}{\pi} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{3}{\pi} \right)} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{3}{\pi} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{3}{\pi} \right)} + \pi$$
, где n - любое целое число

Быстрый ответ

              /3 \
x1 = pi - asin|--|
              \pi/

$$x_{1} = \pi - \operatorname{asin}{\left(\frac{3}{\pi} \right)}$$

         /3 \
x2 = asin|--|
         \pi/

$$x_{2} = \operatorname{asin}{\left(\frac{3}{\pi} \right)}$$

Сумма и произведение корней

сумма

             /3 \       /3 \
0 + pi - asin|--| + asin|--|
             \pi/       \pi/

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

=

pi

$$\pi$$

произведение

  /         /3 \\     /3 \
1*|pi - asin|--||*asin|--|
  \         \pi//     \pi/

$$1 \left(\pi - \operatorname{asin}{\left(\frac{3}{\pi} \right)}\right) \operatorname{asin}{\left(\frac{3}{\pi} \right)}$$

=

/         /3 \\     /3 \
|pi - asin|--||*asin|--|
\         \pi//     \pi/

$$\left(\pi - \operatorname{asin}{\left(\frac{3}{\pi} \right)}\right) \operatorname{asin}{\left(\frac{3}{\pi} \right)}$$

Численный ответ

x1 = 45.8544670867973

x2 = -30.1465038188483

x3 = -86.6951715834646

x4 = -60.9596831352557

x5 = 83.5535789298748

x6 = -10.694200677819

x7 = 20.7217258580789

x8 = 96.119949544234

x9 = -36.4296891260279

x10 = -92.9783568906442

x11 = -5.01376259012998

x12 = 64.7040230083361

x13 = 1.87216993654019

x14 = 390.826911762184

x15 = -29.5437565993577

x16 = 58.4208377011565

x17 = -48.3933125208965

x18 = 8.15535524371978

x19 = -55.2792450475667

x20 = 38.9685345601271

x21 = 39.5712817796177

x22 = -67.2428684424353

x23 = -48.9960597403871

x24 = 76.6676464032046

x25 = -4.4110153706394

x26 = -80.411986276285

x27 = 89.8367642370544

x28 = -17.5801332044892

x29 = 64.1012757888455

x30 = -130.074721514231

x31 = 7.55260802422919

x32 = -11.2969478973096

x33 = 70.3844610960251

x34 = 20.1189786385884

x35 = 77.2703936226952

x36 = -61.5624303547463

x37 = -86.092424363974

x38 = -42.1101272137169

x39 = -67.8456156619258

x40 = -42.7128744332075

x41 = -99.2615421978238

x42 = 14.4385405508994

x43 = 27.0049111652585

x44 = -79.8092390567944

x45 = -92.3756096711536

x46 = 13.8357933314088

x47 = 101.800387631923

x48 = 2169.5711009135

x49 = 57.8180904816659

x50 = 89.2340170175638

x51 = -23.2605712921782

x52 = -98.6587949783332

x53 = -74.1288009691054

x54 = 1.26942271704961

x55 = -16.9773859849986

x56 = 209.217285073467

x57 = 52.1376523939769

x58 = 70.9872083155156

x59 = 32.6853492529475

x60 = 51.5349051744863

x61 = 33.2880964724381

x62 = -54.6764978280761

x63 = 82.9508317103842

x64 = 45.2517198673067

x65 = 95.5172023247434

x66 = -73.5260537496148

x67 = -35.8269419065373

x68 = -23.8633185116687

x69 = 26.402163945768

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