cos(3*x+pi/4)=(-sqrt(3))/2 (уравнение)

Найду корень уравнения: cos(3*x+pi/4)=(-sqrt(3))/2

Вы ввели

                   ___ 
   /      pi\   -\/ 3  
cos|3*x + --| = -------
   \      4 /      2   

$$\cos{\left(3 x + \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$

Подробное решение

Дано уравнение
$$\cos{\left(3 x + \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$3 x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$3 x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
Или
$$3 x + \frac{\pi}{4} = \pi n + \frac{5 \pi}{6}$$
$$3 x + \frac{\pi}{4} = \pi n - \frac{\pi}{6}$$
, где n - любое целое число
Перенесём
$$\frac{\pi}{4}$$
в правую часть ур-ния
с противоположным знаком, итого:
$$3 x = \pi n + \frac{7 \pi}{12}$$
$$3 x = \pi n - \frac{5 \pi}{12}$$
Разделим обе части полученного ур-ния на
$$3$$
получим ответ:
$$x_{1} = \frac{\pi n}{3} + \frac{7 \pi}{36}$$
$$x_{2} = \frac{\pi n}{3} - \frac{5 \pi}{36}$$

Быстрый ответ

     7*pi
x1 = ----
      36 

$$x_{1} = \frac{7 \pi}{36}$$

     11*pi
x2 = -----
       36 

$$x_{2} = \frac{11 \pi}{36}$$

Сумма и произведение корней

сумма

    7*pi   11*pi
0 + ---- + -----
     36      36 

$$\left(0 + \frac{7 \pi}{36}\right) + \frac{11 \pi}{36}$$

=

pi
--
2 

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

произведение

  7*pi 11*pi
1*----*-----
   36    36 

$$\frac{11 \pi}{36} \cdot 1 \cdot \frac{7 \pi}{36}$$

=

     2
77*pi 
------
 1296 

$$\frac{77 \pi^{2}}{1296}$$

Численный ответ

x1 = 28.1870674197084

x2 = 82.2922742315326

x3 = -91.5425192671026

x4 = -53.8434074240251

x5 = -43.3714319120591

x6 = -63.9663170855922

x7 = -66.0607121879854

x8 = 42.4987672860619

x9 = 70.0749694675723

x10 = -82.8158730071309

x11 = -15.7952297305487

x12 = -26.2672052425147

x13 = -93.6369143694958

x14 = -59.7775268808058

x15 = 44.5931623884551

x16 = -41.2770368096659

x17 = 78.1034840267462

x18 = 90.6698546411054

x19 = 40.4043721836687

x20 = 103.236225255465

x21 = 61.6973890579996

x22 = -24.1728101401215

x23 = -45.4658270144523

x24 = -89.4481241647094

x25 = -97.8257045742822

x26 = -22.0784150377283

x27 = -74.4382925975582

x28 = -61.871921983199

x29 = 84.3866693339258

x30 = 76.0090889243531

x31 = -78.9761486527434

x32 = 67.9805743651791

x33 = 49.1310184436404

x34 = 30.2814625221016

x35 = 97.3021057986839

x36 = 55.0651379004211

x37 = -17.8896248329419

x38 = 46.6875574908483

x39 = -7.4176493209759

x40 = 65.8861792627859

x41 = -38.8335758568738

x42 = 13.5263017029561

x43 = 5.14872129338327

x44 = -47.5602221168455

x45 = -11.9555053761612

x46 = -1.48352986419518

x47 = -51.7490123216319

x48 = 72.1693645699655

x49 = 32.0267917740959

x50 = 80.1978791291394

x51 = -70.2495023927718

x52 = 59.6029939556064

x53 = 51.2254135460336

x54 = 88.5754595387122

x55 = -47.2111562664466

x56 = 34.1211868764891

x57 = 19.8094870101356

x58 = 86.481064436319

x59 = -19.9840199353351

x60 = 92.7642497434986

x61 = 26.0926723173152

x62 = 17.7150919077424

x63 = -87.3537290623162

x64 = -99.9200996766754

x65 = -76.8817535503502

x66 = 11.082840750164

x67 = -72.343897495165

x68 = -28.3616003449079

x69 = 23.998277214922

x70 = 21.9038821125288

x71 = 15.6206968053492

x72 = 38.3099770812755

x73 = 2556.1219560083

x74 = 36.2155819788823

x75 = 1626.21053054572

x76 = 63.7917841603927

x77 = -95.731309471889

x78 = 0.610865238198015

x79 = -5.67232006898157

x80 = 2.70526034059121

x81 = -7.76671517137477

x82 = -55.9378025264183

x83 = -68.1551072903786

x84 = -30.4559954473011

x85 = 74.2637596723587

x86 = -3.57792496658838

x87 = -9.86111027376796

x88 = -49.6546172192387

x89 = -85.259333959923

x90 = 57.5085988532132

x91 = -32.8994564000931

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