cos(9x)=sin117 (уравнение)

Найду корень уравнения: cos(9x)=sin117

Вы ввели

cos(9*x) = sin(117)

$$\cos{\left(9 x \right)} = \sin{\left(117 \right)}$$

Подробное решение

Дано уравнение
$$\cos{\left(9 x \right)} = \sin{\left(117 \right)}$$
- это простейшее тригонометрическое ур-ние
Это ур-ние преобразуется в
$$9 x = 2 \pi n + \operatorname{asin}{\left(\sin{\left(117 \right)} \right)}$$
$$9 x = 2 \pi n - \operatorname{asin}{\left(\sin{\left(117 \right)} \right)} + \pi$$
Или
$$9 x = 2 \pi n - 117 + 37 \pi$$
$$9 x = 2 \pi n - 36 \pi + 117$$
, где n - любое целое число
Разделим обе части полученного ур-ния на
$$9$$
получим ответ:
$$x_{1} = \frac{2 \pi n}{9} - 13 + \frac{37 \pi}{9}$$
$$x_{2} = \frac{2 \pi n}{9} - 4 \pi + 13$$

Быстрый ответ

           77*pi
x1 = -13 + -----
             18 

$$x_{1} = -13 + \frac{77 \pi}{18}$$

          73*pi
x2 = 13 - -----
            18 

$$x_{2} = 13 - \frac{73 \pi}{18}$$

Сумма и произведение корней

сумма

          77*pi        73*pi
0 + -13 + ----- + 13 - -----
            18           18 

$$\left(13 - \frac{73 \pi}{18}\right) - \left(13 - \frac{77 \pi}{18}\right)$$

=

2*pi
----
 9  

$$\frac{2 \pi}{9}$$

произведение

  /      77*pi\ /     73*pi\
1*|-13 + -----|*|13 - -----|
  \        18 / \       18 /

$$1 \left(-13 + \frac{77 \pi}{18}\right) \left(13 - \frac{73 \pi}{18}\right)$$

=

              2         
       5621*pi    325*pi
-169 - -------- + ------
         324        3   

$$- \frac{5621 \pi^{2}}{324} - 169 + \frac{325 \pi}{3}$$

Численный ответ

x1 = 30.2787595947439

x2 = 16.3161255787892

x3 = -65.8834763354282

x4 = 6.02408884673819

x5 = 68.6760031386191

x6 = 12.3072741539178

x7 = 47.7320521146872

x8 = -59.7802298081635

x9 = -88.9218224617533

x10 = 22.081117965086

x11 = -91.7143492649443

x12 = -21.9011791851711

x13 = -29.7605666738611

x14 = -100.271868454432

x15 = 38.1381470834339

x16 = 21.9011791851711

x17 = 96.0830782496456

x18 = 23.9955742875643

x19 = -84.0349005561692

x20 = 77.9316540289046

x21 = 264347.129471821

x22 = 40.2325421858271

x23 = 93.9886831472524

x24 = 103.064395257623

x25 = -47.7320521146872

x26 = 82.120444233691

x27 = 88.2236907609556

x28 = 8.11848394913139

x29 = -77.7517152489896

x30 = -27.4862327915529

x31 = -50.0063859969953

x32 = -97.9975345721239

x33 = 44.2413936106985

x34 = 70.2522053201295

x35 = -7.93854516921644

x36 = -20.5049157835756

x37 = -23.9955742875643

x38 = 14.221730476396

x39 = 58.2040276266531

x40 = 33.9493568786475

x41 = 56.2895713041749

x42 = 90.3180858633488

x43 = -52.6189740202713

x44 = -30.2787595947439

x45 = 68.1578102177363

x46 = 3.929693744345

x47 = 28.1843644923507

x48 = 35.165681500328

x49 = -31.8549617762543

x50 = -14.9198621771938

x51 = 72.1666616426078

x52 = -40.0526034059121

x53 = -17.7123889803847

x54 = 26.2699081698724

x55 = -26.2699081698724

x56 = -67.9778714378214

x57 = -57.6858347057703

x58 = -87.7054978400728

x59 = -1.8352986419518

x60 = 349.504885639222

x61 = -89.799892942466

x62 = 0.259096460441394

x63 = -73.7428638241182

x64 = -77.9316540289046

x65 = 50.0063859969953

x66 = 313.022098417825

x67 = 46.5157274930066

x68 = -45.8175957922089

x69 = 63.9690200129499

x70 = 42.1469985083053

x71 = 19.9867228626928

x72 = 97.9975345721239

x73 = 66.0634151153431

x74 = -81.940505453776

x75 = -35.8638132011257

x76 = -3.74975496443005

x77 = 60.2984227290463

x78 = 52.1007810993885

x79 = 80.0260491312978

x80 = -70.0722665402146

x81 = -37.9582083035189

x82 = -61.6946861306418

x83 = 100.271868454432

x84 = -15.7979326579064

x85 = 36.0437519810407

x86 = 74.261056745001

x87 = -95.9031394697307

x88 = -54.0152374218667

x89 = -13.7035375555132

x90 = -72.3466004225227

x91 = -33.7694180987325

x92 = -43.7232006898157

x93 = -51.9208423194736

x94 = -10.0329402716096

x95 = -75.8372589265114

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