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sin6x=1/2 equation

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sin(6*x) = 1/2

\sin{\left(6 x \right)} = \frac{1}{2}

Simplify

Detail solution

Given the equation

\sin{\left(6 x \right)} = \frac{1}{2}

- this is the simplest trigonometric equation
This equation is transformed to

6 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}

6 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi

6 x = 2 \pi n + \frac{\pi}{6}

6 x = 2 \pi n + \frac{5 \pi}{6}

, where n - is a integer
Divide both parts of the equation by

6

we get the answer:

x_{1} = \frac{\pi n}{3} + \frac{\pi}{36}

x_{2} = \frac{\pi n}{3} + \frac{5 \pi}{36}

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

      pi
x_1 = --
      36

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

Simplify

      5*pi
x_2 = ----
       36

x_{2} = \frac{5 \pi}{36}

Simplify

Sum and product of roots [src]

sum

pi   5*pi
-- + ----
36    36

\left(\frac{\pi}{36}\right) + \left(\frac{5 \pi}{36}\right)

pi
--
6

\frac{\pi}{6}

Simplify

product

pi   5*pi
-- * ----
36    36

\left(\frac{\pi}{36}\right) * \left(\frac{5 \pi}{36}\right)

    2
5*pi 
-----
 1296

\frac{5 \pi^{2}}{1296}

Simplify

Numerical answer [src]

x1 = -63.7917841603927

x2 = 16.1442955809475

x3 = -76.0090889243531

x4 = -27.8380015693096

x5 = -5.846852994181

x6 = -34.1211868764891

x7 = -47.7347550420449

x8 = 38.1354441560761

x9 = 82.1177413063332

x10 = 391.04101890933

x11 = 75.4854901487547

x12 = -49.8291501444381

x13 = 22.0784150377283

x14 = 80.02334620394

x15 = -67.9805743651791

x16 = 66.0607121879854

x17 = -51.9235452468313

x18 = -7.9412480965742

x19 = -59.6029939556064

x20 = 60.1265927312047

x21 = 58.0321976288115

x22 = -17.7150919077424

x23 = 70.2495023927718

x24 = 19.9840199353351

x25 = 11.9555053761612

x26 = 77.9289511015468

x27 = -73.9146938219599

x28 = -32.0267917740959

x29 = -87.8773278379145

x30 = -69.7259036171735

x31 = 46.16395871525

x32 = -21.9038821125288

x33 = 24.1728101401215

x34 = 44.0695636128568

x35 = 33.9466539512897

x36 = -65.8861792627859

x37 = 62.9191195343956

x38 = -43.8950306876574

x39 = 6.3704517697793

x40 = 28.3616003449079

x41 = 55.9378025264183

x42 = -91.717052192302

x43 = -78.1034840267462

x44 = 2.18166156499291

x45 = -83.6885376331281

x46 = -3.75245789178781

x47 = 18.2386906833407

x48 = -71.8202987195667

x49 = 36.0410490536829

x50 = 88.0518607631139

x51 = -95.9058423970884

x52 = -45.9894257900506

x53 = 68.1551072903786

x54 = 26.2672052425147

x55 = 94.3350460702935

x56 = 90.1462558655071

x57 = -61.6973890579996

x58 = -37.6118453804778

x59 = 14.0499004785544

x60 = 0.0872664625997165

x61 = -100.094632601875

x62 = 48.2583538176432

x63 = -10.0356431989674

x64 = 62.2209878335978

x65 = 31.8522588488965

x66 = -12.1300383013606

x67 = -85.7829327355213

x68 = -29.9323966717028

x69 = 9.86111027376796

x70 = 41.9751685104636

x71 = 53.4943415736262

x72 = -41.8006355852642

x73 = -81.245076680336

x74 = -56.1123354516177

x75 = -23.998277214922

x76 = -93.8114472946952

x77 = 4.27605666738611

x78 = 84.2121364087264

x79 = 50.3527489200364

x80 = 40.2298392584693

x81 = -54.0179403492245

x82 = -25.7436064669164

x83 = -89.9717229403077

x84 = 99.9200996766754

x85 = 63.9663170855922

x86 = -39.706240482871

x87 = 72.343897495165

x88 = -19.8094870101356

x89 = -98.0002374994816

x90 = -15.6206968053492

x91 = 92.2406509679003

x92 = -2.00712863979348

x93 = 96.7785070230856

x93 = 96.7785070230856

The graph