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13(sin(x)^(2))+sin(2x)-3(cos(x)^(2))=4 equation

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Numerical solution:

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The solution

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      2                      2       
13*sin (x) + sin(2*x) - 3*cos (x) = 4
$$\left(13 \sin^{2}{\left(x \right)} + \sin{\left(2 x \right)}\right) - 3 \cos^{2}{\left(x \right)} = 4$$
Simplify
The graph
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
  pi           /log(130)      /  _____\\                 /log(130)      /  _____\\                    /    5/2\
- -- + -pi + I*|-------- - log\\/ 130 /| + atan(7/9) + I*|-------- - log\\/ 130 /| + atan(7/9) - I*log\(-I)   /
  4            \   2                   /                 \   2                   /
$$- i \log{\left(\left(- i\right)^{\frac{5}{2}} \right)} + \left(\left(- \frac{\pi}{4} + \left(- \pi + \operatorname{atan}{\left(\frac{7}{9} \right)} + i \left(- \log{\left(\sqrt{130} \right)} + \frac{\log{\left(130 \right)}}{2}\right)\right)\right) + \left(\operatorname{atan}{\left(\frac{7}{9} \right)} + i \left(- \log{\left(\sqrt{130} \right)} + \frac{\log{\left(130 \right)}}{2}\right)\right)\right)$$ 
=
              5*pi        /    5/2\       /log(130)      /  _____\\
2*atan(7/9) - ---- - I*log\(-I)   / + 2*I*|-------- - log\\/ 130 /|
               4                          \   2                   /
$$- \frac{5 \pi}{4} + 2 \operatorname{atan}{\left(\frac{7}{9} \right)} - i \log{\left(\left(- i\right)^{\frac{5}{2}} \right)} + 2 i \left(- \log{\left(\sqrt{130} \right)} + \frac{\log{\left(130 \right)}}{2}\right)$$ 
product
-pi  /        /log(130)      /  _____\\            \ /  /log(130)      /  _____\\            \ /      /    5/2\\
----*|-pi + I*|-------- - log\\/ 130 /| + atan(7/9)|*|I*|-------- - log\\/ 130 /| + atan(7/9)|*\-I*log\(-I)   //
 4   \        \   2                   /            / \  \   2                   /            /
$$- i \log{\left(\left(- i\right)^{\frac{5}{2}} \right)} - \frac{\pi}{4} \left(- \pi + \operatorname{atan}{\left(\frac{7}{9} \right)} + i \left(- \log{\left(\sqrt{130} \right)} + \frac{\log{\left(130 \right)}}{2}\right)\right) \left(\operatorname{atan}{\left(\frac{7}{9} \right)} + i \left(- \log{\left(\sqrt{130} \right)} + \frac{\log{\left(130 \right)}}{2}\right)\right)$$ 
=
                                    /    5/2\
pi*I*(-pi + atan(7/9))*atan(7/9)*log\(-I)   /
---------------------------------------------
                      4
$$\frac{i \pi \left(- \pi + \operatorname{atan}{\left(\frac{7}{9} \right)}\right) \log{\left(\left(- i\right)^{\frac{5}{2}} \right)} \operatorname{atan}{\left(\frac{7}{9} \right)}}{4}$$ 
pi*i*(-pi + atan(7/9))*atan(7/9)*log((-i)^(5/2))/4
Rapid solution [src] 
     -pi 
x1 = ----
      4
$$x_{1} = - \frac{\pi}{4}$$ 
             /log(130)      /  _____\\            
x2 = -pi + I*|-------- - log\\/ 130 /| + atan(7/9)
             \   2                   /
$$x_{2} = - \pi + \operatorname{atan}{\left(\frac{7}{9} \right)} + i \left(- \log{\left(\sqrt{130} \right)} + \frac{\log{\left(130 \right)}}{2}\right)$$ 
       /log(130)      /  _____\\            
x3 = I*|-------- - log\\/ 130 /| + atan(7/9)
       \   2                   /
$$x_{3} = \operatorname{atan}{\left(\frac{7}{9} \right)} + i \left(- \log{\left(\sqrt{130} \right)} + \frac{\log{\left(130 \right)}}{2}\right)$$ 
           /    5/2\
x4 = -I*log\(-I)   /
$$x_{4} = - i \log{\left(\left(- i\right)^{\frac{5}{2}} \right)}$$ 
x4 = -i*log((-i)^(5/2))
Numerical answer [src] 
x1 = -65.312402556535
x2 = 82.3424521621853
x3 = 47.7849329726976
x4 = 99.7455667514759
x5 = -25.9181393921158
x6 = -38.484510006475
x7 = -68.4539952101248
x8 = 96.6039740978861
x9 = -71.5955878637146
x10 = 10.0858211296201
x11 = -79.3252145031423
x12 = 22.6521917439792
x13 = 19.5105990903894
x14 = 85.4840448157751
x15 = -35.3429173528852
x16 = -99.8699217460227
x17 = -7239.01487203428
x18 = -10.2101761241668
x19 = 74.6128255227576
x20 = 68.329640215578
x21 = 16.3690064367997
x22 = -85.6083998103219
x23 = 54.0681182798772
x24 = 11.7809724509617
x25 = -77.8787731708941
x26 = -3.92699081698724
x27 = 90.3207887907066
x28 = -82.4668071567321
x29 = -41.6261026600648
x30 = -37.0380686742268
x31 = -27.6132907134575
x32 = -93.5867364388431
x33 = 44.6433403191078
x34 = -19.6349540849362
x35 = -55.8876245957656
x36 = 41.501747665518
x37 = -47.9092879672443
x38 = -32.2013246992954
x39 = -40.1796613278166
x40 = 24.3473430653209
x41 = 3.80263582244048
x42 = -5.6221421383289
x43 = 0.661043168850687
x44 = 8.63937979737193
x45 = -43.3212539814064
x46 = 77.7544181763474
x47 = -33.896476020637
x48 = 60.3513035870568
x49 = -98.174770424681
x50 = 88.6256374693649
x51 = -13.3517687777566
x52 = 55.7632696012188
x53 = -11.9053274455085
x54 = 91.7672301229547
x55 = -76.1836218495525
x56 = -90.4451437852533
x57 = 38.3601550119282
x58 = -60.4756585816035
x59 = 84.037603483527
x60 = -24.4716980598677
x61 = -49.604439288586
x62 = 57.209710933467
x63 = -87.3035511316635
x64 = 76.0592668550057
x65 = -69.9004365423729
x66 = 52.621676947629
x67 = -84.1619584780737
x68 = 63.4928962406466
x69 = -54.1924732744239
x70 = 98.0504154301343
x71 = 69.7760815478261
x72 = 33.7721210260903
x73 = 46.3384916404494
x74 = -2.48054948473911
x75 = -63.6172512351933
x76 = -62.1708099029452
x77 = -91.8915851175014
x78 = 2.35619449019234
x79 = 32.0769697047486
x80 = -461172.574133198
x81 = 40.0553063332699
x82 = 30.6305283725005
x83 = 18.0641577581413
x84 = 62.0464549083984
x85 = 66.6344888942363
x86 = -46.4628466349962
x87 = 21.2057504117311
x88 = -18.1885127526881
x89 = -21.3301054062779
x90 = 25.793784397569
x91 = -57.3340659280137
x91 = -57.3340659280137
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