sin(x)=√5/3 (уравнение)

Найду корень уравнения: sin(x)=√5/3

Вы ввели

           ___
         \/ 5 
sin(x) = -----
           3  

$$\sin{\left(x \right)} = \frac{\sqrt{5}}{3}$$

Подробное решение

Дано уравнение
$$\sin{\left(x \right)} = \frac{\sqrt{5}}{3}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)} + \pi$$
, где n - любое целое число

Быстрый ответ

              /  ___\
              |\/ 5 |
x1 = pi - asin|-----|
              \  3  /

$$x_{1} = \pi - \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}$$

         /  ___\
         |\/ 5 |
x2 = asin|-----|
         \  3  /

$$x_{2} = \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}$$

Сумма и произведение корней

сумма

             /  ___\       /  ___\
             |\/ 5 |       |\/ 5 |
0 + pi - asin|-----| + asin|-----|
             \  3  /       \  3  /

$$\operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)} + \left(0 + \left(\pi - \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}\right)\right)$$

=

pi

$$\pi$$

произведение

  /         /  ___\\     /  ___\
  |         |\/ 5 ||     |\/ 5 |
1*|pi - asin|-----||*asin|-----|
  \         \  3  //     \  3  /

$$1 \left(\pi - \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}\right) \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}$$

=

/         /  ___\\     /  ___\
|         |\/ 5 ||     |\/ 5 |
|pi - asin|-----||*asin|-----|
\         \  3  //     \  3  /

$$\left(\pi - \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}\right) \operatorname{asin}{\left(\frac{\sqrt{5}}{3} \right)}$$

Численный ответ

x1 = 33.7164505189198

x2 = 13.4074392849271

x3 = -36.8580431725096

x4 = 69.9561070495434

x5 = -897.654430256113

x6 = 76.239292356723

x7 = 65.1323770548177

x8 = -47.9649584744148

x9 = -79.3808850103128

x10 = -60.531329088774

x11 = -85.6640703174923

x12 = -80.8403403227667

x13 = 245.885295650572

x14 = 19.6906245921067

x15 = 52.5660064404586

x16 = -73.0976997031332

x17 = -3.98266132415772

x18 = -55.7075990940483

x19 = 102.831488897895

x20 = 63.6729217423638

x21 = -98.2304409318515

x22 = 3475.44254354088

x23 = 96.5483035907157

x24 = 101.372033585441

x25 = -24.2916725581504

x26 = 57.3897364351842

x27 = 88.8056629710821

x28 = -87.1235256299463

x29 = 77.6987476691769

x30 = -489.24738528944

x31 = 2.30052398302186

x32 = 71.4155623619973

x33 = -16.5490319385169

x34 = -10.2658466313373

x35 = -68.2739697084075

x36 = 83.9819329763565

x37 = -74.5571550155871

x38 = 25.9738098992863

x39 = -18.0084872509708

x40 = -66.8145143959536

x41 = -43.1412284796892

x42 = 8.58370929020145

x43 = 7.12425397774752

x44 = 113.9384041998

x45 = 90.2651182835361

x46 = -91.9472556246719

x47 = 82.5224776639025

x48 = 58.8491917476381

x49 = -93.4067109371259

x50 = -61.9907844012279

x51 = -41.6817731672352

x52 = -118.539452165844

x53 = 95.0888482782617

x54 = -311.858741375957

x55 = 39.9996358260994

x56 = -217.610961768264

x57 = -35.3985878600557

x58 = 32.2569952064659

x59 = -99.6898962443055

x60 = 0.84106867056793

x61 = -29.1154025528761

x62 = -22.8322172456965

x63 = -11.7253019437912

x64 = 17739.732646151

x65 = 21.1500799045606

x66 = -54.2481437815944

x67 = 46.282821133279

x68 = -30.57485786533

x69 = -5.44211663661166

x70 = 38.5401805136454

x71 = 27.4332652117402

x72 = -49.4244137868688

x73 = 44.823365820825

x74 = 51.1065511280046

x75 = 14.866894597381

График