cos(−10π−4x)=–√2/2 (уравнение)

Найду корень уравнения: cos(−10π−4x)=–√2/2

Вы ввели

                       ___ 
                    -\/ 2  
cos(-10*pi - 4*x) = -------
                       2   

$$\cos{\left(- 4 x - 10 \pi \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(- 4 x - 10 \pi \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$4 x = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$4 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}$$
Или
$$4 x = \pi n + \frac{3 \pi}{4}$$
$$4 x = \pi n - \frac{\pi}{4}$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$4$$
получим ответ:
$$x_{1} = \frac{\pi n}{4} + \frac{3 \pi}{16}$$
$$x_{2} = \frac{\pi n}{4} - \frac{\pi}{16}$$

Быстрый ответ

     3*pi
x1 = ----
      16 

$$x_{1} = \frac{3 \pi}{16}$$

     5*pi
x2 = ----
      16 

$$x_{2} = \frac{5 \pi}{16}$$

Сумма и произведение корней

сумма

3*pi   5*pi
---- + ----
 16     16 

$$\frac{3 \pi}{16} + \frac{5 \pi}{16}$$

=

pi
--
2 

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

произведение

3*pi 5*pi
----*----
 16   16 

$$\frac{3 \pi}{16} \frac{5 \pi}{16}$$

=

     2
15*pi 
------
 256  

$$\frac{15 \pi^{2}}{256}$$

Численный ответ

x1 = -49.6764338348886

x2 = 10.0138265833175

x3 = -106.225101599505

x4 = -19.8313036257856

x5 = -13.548118318606

x6 = -47.712938426395

x7 = -32.004975158446

x8 = 66.5624943479338

x9 = -88.946342004761

x10 = 85.8047493511712

x11 = 88.5536429230623

x12 = -55.5669200603695

x13 = 2.15984494934298

x14 = 82.2704576158827

x15 = -61.8501053675491

x16 = -38.2881604656256

x17 = -21.0094008708817

x18 = -91.6952355766521

x19 = -93.6587309851457

x20 = -39.8589567924205

x21 = 62.2428044492478

x22 = 48.1056375080937

x23 = 52.4253274067797

x24 = 90.1244392498572

x25 = -5.6941366846315

x26 = 26.1144889329652

x27 = -27.6852852597601

x28 = -11.9773219918111

x29 = 19.8313036257856

x30 = -55.9596191420682

x31 = -33.9684705669396

x32 = 16.2970118904971

x33 = -85.8047493511712

x34 = -99.9419162923253

x35 = 22.9728962793754

x36 = 32.004975158446

x37 = 419.991667876785

x38 = 18.2605072989907

x39 = -68.1332906747286

x40 = -90.1244392498572

x41 = 60.2793090407542

x42 = -75.9872723087031

x43 = -60.2793090407542

x44 = -57.5304154688631

x45 = 57.1377163871644

x46 = 99.9419162923253

x47 = -63.8136007760427

x48 = -77.9507677171967

x49 = -79.5215640439916

x50 = 97.9784208838317

x51 = -10.0138265833175

x52 = 70.0967860832223

x53 = -82.2704576158827

x54 = 84.2339530243763

x55 = 5.6941366846315

x56 = 74.4164759819082

x57 = -24.1509935244715

x58 = 63.8136007760427

x59 = -2.15984494934298

x60 = 46.1421420996001

x61 = 53.9961237335746

x62 = -41.8224522009141

x63 = -25.7217898512664

x64 = -43.0005494460103

x65 = -69.7040870015235

x66 = -71.6675824100172

x67 = -35.5392668937345

x68 = 71.2748833283184

x69 = 33.9684705669396

x70 = -16.2970118904971

x71 = -46.1421420996001

x72 = -83.8412539426776

x73 = -17.8678082172919

x74 = 38.2881604656256

x75 = -3.73064127613788

x76 = 55.9596191420682

x77 = 27.6852852597601

x78 = 41.8224522009141

x79 = 0.98174770424681

x80 = -97.9784208838317

x81 = 96.4076245570368

x82 = 4.1233403578366

x83 = 11.9773219918111

x84 = -64.9916980211388

x85 = 68.1332906747286

x86 = 24.1509935244715

x87 = 77.9507677171967

x88 = -53.9961237335746

x89 = 30.4341788316511

x90 = 93.266031903447

x91 = 8.44303025652257

x92 = 92.0879346583508

x93 = 75.9872723087031

x94 = 40.2516558741192

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