sin(x)=pi/4 (уравнение)

Найду корень уравнения: sin(x)=pi/4

Вы ввели

         pi
sin(x) = --
         4 

$$\sin{\left(x \right)} = \frac{\pi}{4}$$

Подробное решение

Дано уравнение
$$\sin{\left(x \right)} = \frac{\pi}{4}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{\pi}{4} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{\pi}{4} \right)} + \pi$$
Или
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{\pi}{4} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{\pi}{4} \right)} + \pi$$
, где n - любое целое число

Быстрый ответ

              /pi\
x1 = pi - asin|--|
              \4 /

$$x_{1} = \pi - \operatorname{asin}{\left(\frac{\pi}{4} \right)}$$

         /pi\
x2 = asin|--|
         \4 /

$$x_{2} = \operatorname{asin}{\left(\frac{\pi}{4} \right)}$$

Сумма и произведение корней

сумма

             /pi\       /pi\
0 + pi - asin|--| + asin|--|
             \4 /       \4 /

$$\operatorname{asin}{\left(\frac{\pi}{4} \right)} + \left(0 + \left(\pi - \operatorname{asin}{\left(\frac{\pi}{4} \right)}\right)\right)$$

=

pi

$$\pi$$

произведение

  /         /pi\\     /pi\
1*|pi - asin|--||*asin|--|
  \         \4 //     \4 /

$$1 \left(\pi - \operatorname{asin}{\left(\frac{\pi}{4} \right)}\right) \operatorname{asin}{\left(\frac{\pi}{4} \right)}$$

=

/         /pi\\     /pi\
|pi - asin|--||*asin|--|
\         \4 //     \4 /

$$\left(\pi - \operatorname{asin}{\left(\frac{\pi}{4} \right)}\right) \operatorname{asin}{\left(\frac{\pi}{4} \right)}$$

Численный ответ

x1 = 76.3015627969215

x2 = -48.0272289146134

x3 = 51.1688215682032

x4 = 27.3709947715416

x5 = -4.04493176435631

x6 = -87.0612551897477

x7 = 26.0360803394849

x8 = 14.8046241571825

x9 = 57.4520068753828

x10 = 44.8856362610236

x11 = -30.5125874251314

x12 = -80.7780698825681

x13 = 2.23825354282328

x14 = -61.9285139610294

x15 = -85.7263407576909

x16 = -55.6453286538498

x17 = 77.6364772289783

x18 = -17.9462168107722

x19 = 38.602450953844

x20 = -93.3444404969273

x21 = -43.0789580394906

x22 = 101.43430402564

x23 = -98.2927113720501

x24 = 8.52143885000287

x25 = -49.3621433466702

x26 = 32.3192656466644

x27 = -54.310414221793

x28 = -68.2116992682089

x29 = 13.4697097251257

x30 = 88.8679334112807

x31 = 0.903339110766513

x32 = -35.4608583002542

x33 = 39.9373653859008

x34 = 58.7869213074396

x35 = 21.087809464362

x36 = 96.4860331505171

x37 = -66.8767848361522

x38 = -25620.5914291355

x39 = -41.7440436074338

x40 = -92.0095260648705

x41 = -387.319235502311

x42 = -22.8944876858951

x43 = 46.2205506930804

x44 = -193.875405411801

x45 = -24.2294021179518

x46 = 52.50373600026

x47 = 63.7351921825624

x48 = 70.018377489742

x49 = -60.5935995289726

x50 = 95.1511187184603

x51 = -11.6630315035927

x52 = 83.9196625361579

x53 = 19.7528950323053

x54 = 90.2028478433375

x55 = -10.3281170715359

x56 = 33.6541800787212

x57 = 65.0701066146191

x58 = 71.3532919217987

x59 = -73.1599701433318

x60 = -79.4431554505113

x61 = 82.5847481041011

x62 = -74.4948845753885

x63 = -16.6113023787155

x64 = -5.37984619641307

x65 = 7.1865244179461

x66 = -36.795772732311

x67 = -29.1776729930747

x68 = -99.6276258041069

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