sin(x)=√2/3 (уравнение)

Найду корень уравнения: sin(x)=√2/3

Вы ввели

           ___
         \/ 2 
sin(x) = -----
           3  

$$\sin{\left(x \right)} = \frac{\sqrt{2}}{3}$$

Подробное решение

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

Быстрый ответ

              /  ___\
              |\/ 2 |
x1 = pi - asin|-----|
              \  3  /

$$x_{1} = \pi - \operatorname{asin}{\left(\frac{\sqrt{2}}{3} \right)}$$

         /  ___\
         |\/ 2 |
x2 = asin|-----|
         \  3  /

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

Сумма и произведение корней

сумма

             /  ___\       /  ___\
             |\/ 2 |       |\/ 2 |
0 + pi - asin|-----| + asin|-----|
             \  3  /       \  3  /

$$\operatorname{asin}{\left(\frac{\sqrt{2}}{3} \right)} + \left(0 + \left(\pi - \operatorname{asin}{\left(\frac{\sqrt{2}}{3} \right)}\right)\right)$$

=

pi

$$\pi$$

произведение

  /         /  ___\\     /  ___\
  |         |\/ 2 ||     |\/ 2 |
1*|pi - asin|-----||*asin|-----|
  \         \  3  //     \  3  /

$$1 \left(\pi - \operatorname{asin}{\left(\frac{\sqrt{2}}{3} \right)}\right) \operatorname{asin}{\left(\frac{\sqrt{2}}{3} \right)}$$

=

/         /  ___\\     /  ___\
|         |\/ 2 ||     |\/ 2 |
|pi - asin|-----||*asin|-----|
\         \  3  //     \  3  /

$$\left(\pi - \operatorname{asin}{\left(\frac{\sqrt{2}}{3} \right)}\right) \operatorname{asin}{\left(\frac{\sqrt{2}}{3} \right)}$$

Численный ответ

x1 = -154.428922704189

x2 = -49.7745997791474

x3 = -37.2082291647882

x4 = -35.048401867777

x5 = -22.4820312534179

x6 = 19.3404385998281

x7 = 82.1722916716239

x8 = -97.8802549395729

x9 = -194.287861844278

x10 = 59.1993777399168

x11 = -85.3138843252137

x12 = -16.1988459462383

x13 = -87.4737116222249

x14 = 34.0666365111984

x15 = 69.6059210572648

x16 = 25.6236239070077

x17 = -188.004676537098

x18 = 94.7386622859831

x19 = -62.3409703935066

x20 = -9.91566063905869

x21 = 75.8891063644443

x22 = -3.6324753318791

x23 = 88.4554769788035

x24 = 63.3227357500852

x25 = -93.7568969294045

x26 = -2669.86287287304

x27 = -100.040082236584

x28 = 71.7657483542759

x29 = -24.641858550429

x30 = -91.5970696323933

x31 = -28.7652165605975

x32 = 52.9161924327372

x33 = 6.7740679854689

x34 = 78.0489336614555

x35 = 21.5002658968392

x36 = -56.057785086327

x37 = -68.6241557006861

x38 = 96.8984895829943

x39 = 84.3321189686351

x40 = 31.9068092141872

x41 = 65.4825630470963

x42 = -47.6147724821362

x43 = -74.9073410078657

x44 = 15.2170805896597

x45 = 2.65070997530048

x46 = -18.3586732432494

x47 = 90.6153042758147

x48 = 50.756365135726

x49 = -60.1811430964954

x50 = -30.9250438576086

x51 = -66.464328403675

x52 = 13.0572532926485

x53 = -53.8979577893158

x54 = -43.4914144719678

x55 = 38.1899945213668

x56 = 46.6330071255576

x57 = -79.0306990180341

x58 = 57.0395504429056

x59 = -5.79230262889028

x60 = -41.3315871749566

x61 = 0.490882678289311

x62 = -72.7475137108546

x63 = 8.93389528248007

x64 = 27.7834512040188

x65 = -12.0754879360699

x66 = -81.1905263150453

x67 = 101.021847593163

x68 = 40.349821818378

x69 = 44.4731798285464

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