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sin6x+sin2x+2sin^2x=1 equation

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Numerical solution:

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The solution

You have entered [src] 
                           2       
sin(6*x) + sin(2*x) + 2*sin (x) = 1
(sin⁡(2x)+sin⁡(6x))+2sin⁡2(x)=1\left(\sin{\left(2 x \right)} + \sin{\left(6 x \right)}\right) + 2 \sin^{2}{\left(x \right)} = 1
Simplify
The graph
0-80-60-40-2020406080-1001005-5
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
  3*pi   pi   pi   3*pi
- ---- - -- + -- + ----
   4     4    4     4
((−3π4−π4)+π4)+3π4\left(\left(- \frac{3 \pi}{4} - \frac{\pi}{4}\right) + \frac{\pi}{4}\right) + \frac{3 \pi}{4} 
=
0
00 
product
-3*pi -pi  pi 3*pi
-----*----*--*----
  4    4   4   4
3π4π4⋅−3π4(−π4)\frac{3 \pi}{4} \frac{\pi}{4} \cdot - \frac{3 \pi}{4} \left(- \frac{\pi}{4}\right) 
=
    4
9*pi 
-----
 256
9π4256\frac{9 \pi^{4}}{256} 
9*pi^4/256
Rapid solution [src] 
     -3*pi
x1 = -----
       4
x1=−3π4x_{1} = - \frac{3 \pi}{4} 
     -pi 
x2 = ----
      4
x2=−π4x_{2} = - \frac{\pi}{4} 
     pi
x3 = --
     4
x3=π4x_{3} = \frac{\pi}{4} 
     3*pi
x4 = ----
      4
x4=3π4x_{4} = \frac{3 \pi}{4} 
x4 = 3*pi/4
Numerical answer [src] 
x1 = -87.8336946066146
x2 = -72.1257313386657
x3 = 10.2101761241668
x4 = -95.6876762405891
x5 = 7.98488132787406
x6 = 82.3359074628325
x7 = -57.9885643975116
x8 = -46.469391334349
x9 = -79.9797129726402
x10 = 88.0954939944138
x11 = -43.8513974563575
x12 = 3.92699081698724
x13 = -15.5770635740494
x14 = 2.22529479629277
x15 = 99.7455667514759
x16 = -3.92699081698724
x17 = 66.1043454192852
x18 = -11.7809724509617
x19 = -54.1924732744239
x20 = 88.6190927700121
x21 = -7.72308194007491
x22 = 46.2075919465499
x23 = 69.9004365423729
x24 = -10.2101761241668
x25 = -76.1836218495525
x26 = -25.9181393921158
x27 = 33.1176225565924
x28 = 77.7544181763474
x29 = -70.0313362362725
x30 = -33.7721210260903
x31 = 14.2680666350536
x32 = -59.5593607243065
x33 = 36.2592152101822
x34 = -29.7142305152035
x35 = -41.7570023539643
x36 = -18.0641577581413
x37 = 50.3963821513363
x38 = 3.27249234748937
x39 = -90.9752872602044
x40 = 29.9760299030026
x41 = -25.0018415348188
x42 = -85.7392995042214
x43 = -14.0062672472545
x44 = -77.7544181763474
x45 = -63.7481509290929
x46 = 40.0553063332699
x47 = 94.3786793015934
x48 = -5.49778714378214
x49 = -35.997415822383
x50 = -98.174770424681
x51 = 68.1987405216784
x52 = -65.8425460314861
x53 = -62.0464549083984
x54 = 80.2415123604393
x55 = -92.022484811401
x56 = 109.039445018346
x57 = 53.5379748049261
x58 = 84.037603483527
x59 = 60.3447588877039
x60 = -99.7455667514759
x61 = 38.3536103125754
x62 = -68.9841386850759
x63 = 33.7721210260903
x64 = -73.6965276654606
x65 = 72.3875307264648
x66 = -32.2013246992954
x67 = 11.7809724509617
x68 = -27.4889357189107
x69 = 24.2164433714213
x70 = 47.9092879672443
x71 = -21.860248881229
x72 = -82.5977068506317
x73 = 18.0641577581413
x74 = -55.7632696012188
x75 = 51.9671784781312
x76 = -62.7009533778963
x77 = 58.2503637853107
x78 = -19.7658537788358
x79 = -94.1168799137942
x80 = 55.7632696012188
x81 = 62.0464549083984
x82 = 97.5202719551832
x83 = -37.5682121491779
x84 = 73.9583270532597
x85 = -51.705379090332
x86 = -84.037603483527
x87 = 76.1836218495525
x88 = 6.41408500107916
x89 = 0.130899693899575
x90 = -2.35619449019234
x91 = 22.1220482690281
x92 = 25.9181393921158
x93 = 44.1131968441567
x94 = -40.0553063332699
x95 = 28.4052335762077
x96 = 90.189889096807
x97 = 16.4933614313464
x98 = -48.5637864367422
x99 = 95.9494756283883
x100 = 16.3624617374468
x101 = 91.8915851175014
x101 = 91.8915851175014
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