cos(2x)=1/4 (уравнение)

Найду корень уравнения: cos(2x)=1/4

Вы ввели

cos(2*x) = 1/4

$$\cos{\left(2 x \right)} = \frac{1}{4}$$

Подробное решение

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

Быстрый ответ

          acos(1/4)
x1 = pi - ---------
              2    

$$x_{1} = \pi - \frac{\operatorname{acos}{\left(\frac{1}{4} \right)}}{2}$$

     acos(1/4)
x2 = ---------
         2    

$$x_{2} = \frac{\operatorname{acos}{\left(\frac{1}{4} \right)}}{2}$$

Сумма и произведение корней

сумма

         acos(1/4)   acos(1/4)
0 + pi - --------- + ---------
             2           2    

$$\frac{\operatorname{acos}{\left(\frac{1}{4} \right)}}{2} + \left(0 + \left(\pi - \frac{\operatorname{acos}{\left(\frac{1}{4} \right)}}{2}\right)\right)$$

=

pi

$$\pi$$

произведение

  /     acos(1/4)\ acos(1/4)
1*|pi - ---------|*---------
  \         2    /     2    

$$1 \left(\pi - \frac{\operatorname{acos}{\left(\frac{1}{4} \right)}}{2}\right) \frac{\operatorname{acos}{\left(\frac{1}{4} \right)}}{2}$$

=

(-acos(1/4) + 2*pi)*acos(1/4)
-----------------------------
              4              

$$\frac{\left(- \operatorname{acos}{\left(\frac{1}{4} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{1}{4} \right)}}{4}$$

Численный ответ

x1 = 74.7391656503286

x2 = 129.464356833008

x3 = -35.2165772253141

x4 = 99.871906879047

x5 = -77.8807583039184

x6 = -63.4909111076223

x7 = 3.8006506894162

x8 = -65.3143876895593

x9 = 2.48253461776338

x10 = -62.1727950359695

x11 = 11.9073125785328

x12 = -93.5887215718674

x13 = -84.163943611098

x14 = -3.8006506894162

x15 = 27.6152758464817

x16 = -19.5086139573652

x17 = 11748.89746639

x18 = 55.8896097287899

x19 = 98.04843029711

x20 = -46.4648317680205

x21 = 16.3670213037754

x22 = -47.7829478396733

x23 = -98.04843029711

x24 = -33.8984611536613

x25 = -76.0572817219814

x26 = -99.871906879047

x27 = 25.7917992645448

x28 = 71.5975729967388

x29 = -25.7917992645448

x30 = -18.1904978857124

x31 = 90.4471289182776

x32 = 24.4736831928919

x33 = -43.3232391144307

x34 = -41.4997625324937

x35 = 22.650206610955

x36 = 46.4648317680205

x37 = 19.5086139573652

x38 = -40.1816464608409

x39 = 0.659058035826409

x40 = 66.6325037612121

x41 = 85.4820596827508

x42 = 93.5887215718674

x43 = -49.6064244216103

x44 = -16.3670213037754

x45 = 33.8984611536613

x46 = -71.5975729967388

x47 = 5.62412727135318

x48 = -11.9073125785328

x49 = -24.4736831928919

x50 = -85.4820596827508

x51 = 76.0572817219814

x52 = -57.2077258004427

x53 = -10.0838359965958

x54 = 38.3581698789039

x55 = 82.340467029161

x56 = -82.340467029161

x57 = -69.7740964148019

x58 = 18.1904978857124

x59 = 60.3493184540325

x60 = 77.8807583039184

x61 = -55.8896097287899

x62 = 63.4909111076223

x63 = -5.62412727135318

x64 = 68.455980343149

x65 = -79.1988743755712

x66 = 30.7568685000715

x67 = 49.6064244216103

x68 = 8.76571992494297

x69 = 84.163943611098

x70 = -2.48253461776338

x71 = -54.0661331468529

x72 = 62.1727950359695

x73 = 88.6236523363406

x74 = 91.7652449899304

x75 = 10.0838359965958

x76 = 41.4997625324937

x77 = 44.6413551860835

x78 = 69.7740964148019

x79 = -90.4471289182776

x80 = -68.455980343149

x81 = -38.3581698789039

x82 = -13.2254286501856

x83 = 52.7480170752001

x84 = -60.3493184540325

x85 = -21.3320905393021

x86 = 96.7303142254572

x87 = 40.1816464608409

x88 = 32.0749845717243

x89 = -101.1900229507

x90 = -27.6152758464817

x91 = 54.0661331468529

x92 = -32.0749845717243

x93 = -91.7652449899304

x94 = 47.7829478396733

x95 = 113.756393565059

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