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3sin2x+8sin^2(x)=7 equation

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Numerical solution:

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The solution

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                  2       
3*sin(2*x) + 8*sin (x) = 7
$$8 \sin^{2}{\left(x \right)} + 3 \sin{\left(2 x \right)} = 7$$
Simplify
The graph
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
     pi
x1 = --
     4
$$x_{1} = \frac{\pi}{4}$$ 
           /   ___\
x2 = -I*log\-\/ I /
$$x_{2} = - i \log{\left(- \sqrt{i} \right)}$$ 
           /   ___           \
           |-\/ 2 *(1 - 7*I) |
x3 = -I*log|-----------------|
           \        10       /
$$x_{3} = - i \log{\left(- \frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)}$$ 
           /  ___          \
           |\/ 2 *(1 - 7*I)|
x4 = -I*log|---------------|
           \       10      /
$$x_{4} = - i \log{\left(\frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)}$$ 
x4 = -i*log(sqrt(2)*(1 - 7*i)/10)
Sum and product of roots [src] 
sum
                          /   ___           \        /  ___          \
pi        /   ___\        |-\/ 2 *(1 - 7*I) |        |\/ 2 *(1 - 7*I)|
-- - I*log\-\/ I / - I*log|-----------------| - I*log|---------------|
4                         \        10       /        \       10      /
$$- i \log{\left(\frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)} + \left(\left(- i \log{\left(- \sqrt{i} \right)} + \frac{\pi}{4}\right) - i \log{\left(- \frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)}\right)$$ 
=
                          /   ___           \        /  ___          \
pi        /   ___\        |-\/ 2 *(1 - 7*I) |        |\/ 2 *(1 - 7*I)|
-- - I*log\-\/ I / - I*log|-----------------| - I*log|---------------|
4                         \        10       /        \       10      /
$$- i \log{\left(- \sqrt{i} \right)} - i \log{\left(\frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)} + \frac{\pi}{4} - i \log{\left(- \frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)}$$ 
product
                    /      /   ___           \\ /      /  ___          \\
pi /      /   ___\\ |      |-\/ 2 *(1 - 7*I) || |      |\/ 2 *(1 - 7*I)||
--*\-I*log\-\/ I //*|-I*log|-----------------||*|-I*log|---------------||
4                   \      \        10       // \      \       10      //
$$- i \log{\left(\frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)} - i \log{\left(- \frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)} \frac{\pi}{4} \left(- i \log{\left(- \sqrt{i} \right)}\right)$$ 
=
                    /  ___          \    /  ___           \
        /   ___\    |\/ 2 *(1 - 7*I)|    |\/ 2 *(-1 + 7*I)|
pi*I*log\-\/ I /*log|---------------|*log|----------------|
                    \       10      /    \       10       /
-----------------------------------------------------------
                             4
$$\frac{i \pi \log{\left(- \sqrt{i} \right)} \log{\left(\frac{\sqrt{2} \left(-1 + 7 i\right)}{10} \right)} \log{\left(\frac{\sqrt{2} \left(1 - 7 i\right)}{10} \right)}}{4}$$ 
pi*i*log(-sqrt(i))*log(sqrt(2)*(1 - 7*i)/10)*log(sqrt(2)*(-1 + 7*i)/10)/4
Numerical answer [src] 
x1 = -5.49778714378214
x2 = -40.0553063332699
x3 = -43.1968989868597
x4 = 17.420656649348
x5 = 83.3941023747337
x6 = 1.71269338139906
x7 = 13.3517687777566
x8 = 54.1924732744239
x9 = -115.453530019425
x10 = -1.42889927219073
x11 = 29.9870272637072
x12 = -51.6943817296274
x13 = 44.7676953136546
x14 = 45.6949905316562
x15 = 67.6861391067847
x16 = 86.5356950283235
x17 = 4.85428603498885
x18 = -18.0641577581413
x19 = 55.1197684924255
x20 = -4.57049192578053
x21 = 95.9604729890929
x22 = -26.5616405009091
x23 = 36.2702125708868
x24 = 104.457955731861
x25 = -84.037603483527
x26 = -64.2607523439866
x27 = 89.6772876819133
x28 = -27.4889357189107
x29 = 51.9781758388358
x30 = -55.7632696012188
x31 = -79.9687156119356
x32 = -65.1880475619882
x33 = -21625.9384291487
x34 = 39.4118052244766
x35 = 23.7038419565276
x36 = 25.9181393921158
x37 = -77.7544181763474
x38 = 47.9092879672443
x39 = 91.8915851175014
x40 = -20.2784551937295
x41 = -92.5350862262947
x42 = -73.685530304756
x43 = 38.484510006475
x44 = -95.6766788798845
x45 = -903.065990852461
x46 = 73.9693244139643
x47 = 66.7588438887831
x48 = 0.785398163397448
x49 = 88.7499924639117
x50 = -21.2057504117311
x51 = 98.174770424681
x52 = 10.2101761241668
x53 = 14.2790639957582
x54 = -86.2519009191151
x55 = -42.269603768858
x56 = -139.658976030142
x57 = -13768.8152025207
x58 = 58.2613611460153
x59 = -49.4800842940392
x60 = 22.776546738526
x61 = 60.4756585816035
x62 = -7.71208457937032
x63 = 7.99587868857865
x64 = -70.5439376511662
x65 = -99.7455667514759
x66 = -62.0464549083984
x67 = -71.4712328691678
x68 = 76.1836218495525
x69 = -57.977567036807
x70 = 3.92699081698724
x71 = -33.7721210260903
x72 = -87.1791961371168
x73 = -35.9864184616785
x74 = 69.9004365423729
x75 = -11.7809724509617
x76 = 80.2525097211439
x77 = -13.9952698865499
x78 = -48.5527890760376
x79 = 82.4668071567321
x80 = -76.8271229583458
x81 = 61.4029537996051
x82 = -93.4623814442964
x83 = 32.2013246992954
x84 = -29.7032331544989
x85 = 64.5445464531949
x86 = 16.4933614313464
x86 = 16.4933614313464
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