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✓8+10cosx=2sinx equation

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Numerical solution:

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The solution

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\/ 8  + 10*cos(x) = 2*sin(x)
$$10 \cos{\left(x \right)} + \sqrt{8} = 2 \sin{\left(x \right)}$$
Simplify
The graph
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
            /       ___       ___        ___\
            |5    \/ 2    4*\/ 3    10*\/ 6 |
x1 = -2*atan|-- + ----- + ------- + --------|
            \23     23       23        23   /
$$x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{2}}{23} + \frac{5}{23} + \frac{4 \sqrt{3}}{23} + \frac{10 \sqrt{6}}{23} \right)}$$ 
            /          ___       ___     ___\
            |5    10*\/ 6    4*\/ 3    \/ 2 |
x2 = -2*atan|-- - -------- - ------- + -----|
            \23      23         23       23 /
$$x_{2} = - 2 \operatorname{atan}{\left(- \frac{10 \sqrt{6}}{23} - \frac{4 \sqrt{3}}{23} + \frac{\sqrt{2}}{23} + \frac{5}{23} \right)}$$ 
x2 = -2*atan(-10*sqrt(6)/23 - 4*sqrt(3)/23 + sqrt(2)/23 + 5/23)
Sum and product of roots [src] 
sum
        /       ___       ___        ___\         /          ___       ___     ___\
        |5    \/ 2    4*\/ 3    10*\/ 6 |         |5    10*\/ 6    4*\/ 3    \/ 2 |
- 2*atan|-- + ----- + ------- + --------| - 2*atan|-- - -------- - ------- + -----|
        \23     23       23        23   /         \23      23         23       23 /
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{2}}{23} + \frac{5}{23} + \frac{4 \sqrt{3}}{23} + \frac{10 \sqrt{6}}{23} \right)} - 2 \operatorname{atan}{\left(- \frac{10 \sqrt{6}}{23} - \frac{4 \sqrt{3}}{23} + \frac{\sqrt{2}}{23} + \frac{5}{23} \right)}$$ 
=
        /          ___       ___     ___\         /       ___       ___        ___\
        |5    10*\/ 6    4*\/ 3    \/ 2 |         |5    \/ 2    4*\/ 3    10*\/ 6 |
- 2*atan|-- - -------- - ------- + -----| - 2*atan|-- + ----- + ------- + --------|
        \23      23         23       23 /         \23     23       23        23   /
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{2}}{23} + \frac{5}{23} + \frac{4 \sqrt{3}}{23} + \frac{10 \sqrt{6}}{23} \right)} - 2 \operatorname{atan}{\left(- \frac{10 \sqrt{6}}{23} - \frac{4 \sqrt{3}}{23} + \frac{\sqrt{2}}{23} + \frac{5}{23} \right)}$$ 
product
       /       ___       ___        ___\        /          ___       ___     ___\
       |5    \/ 2    4*\/ 3    10*\/ 6 |        |5    10*\/ 6    4*\/ 3    \/ 2 |
-2*atan|-- + ----- + ------- + --------|*-2*atan|-- - -------- - ------- + -----|
       \23     23       23        23   /        \23      23         23       23 /
$$- 2 \operatorname{atan}{\left(\frac{\sqrt{2}}{23} + \frac{5}{23} + \frac{4 \sqrt{3}}{23} + \frac{10 \sqrt{6}}{23} \right)} \left(- 2 \operatorname{atan}{\left(- \frac{10 \sqrt{6}}{23} - \frac{4 \sqrt{3}}{23} + \frac{\sqrt{2}}{23} + \frac{5}{23} \right)}\right)$$ 
=
      /          ___       ___     ___\     /       ___       ___        ___\
      |5    10*\/ 6    4*\/ 3    \/ 2 |     |5    \/ 2    4*\/ 3    10*\/ 6 |
4*atan|-- - -------- - ------- + -----|*atan|-- + ----- + ------- + --------|
      \23      23         23       23 /     \23     23       23        23   /
$$4 \operatorname{atan}{\left(\frac{\sqrt{2}}{23} + \frac{5}{23} + \frac{4 \sqrt{3}}{23} + \frac{10 \sqrt{6}}{23} \right)} \operatorname{atan}{\left(- \frac{10 \sqrt{6}}{23} - \frac{4 \sqrt{3}}{23} + \frac{\sqrt{2}}{23} + \frac{5}{23} \right)}$$ 
4*atan(5/23 - 10*sqrt(6)/23 - 4*sqrt(3)/23 + sqrt(2)/23)*atan(5/23 + sqrt(2)/23 + 4*sqrt(3)/23 + 10*sqrt(6)/23)
Numerical answer [src] 
x1 = -8.33241209532718
x2 = 33.0703622043458
x3 = -77.4474504743026
x4 = -39.7483386312251
x5 = -46.0315239384047
x6 = 14.220806282807
x7 = 1.65443566844783
x8 = 20.5039915899866
x9 = -83.7306357814822
x10 = -20.8987827096864
x11 = 60.7826262836483
x12 = 67.0658115908279
x13 = 92.1985528195462
x14 = -98.8765292464255
x15 = -42.3278614818093
x16 = -80.0269733248868
x17 = 39.3535475115253
x18 = -36.0446761746297
x19 = -10.9119349459113
x20 = 85.9153675123666
x21 = 2014.8532568165
x22 = -71.164265167123
x23 = -92.593343939246
x24 = 35.6498850549299
x25 = -54.8942320961685
x26 = -2.04922678814759
x27 = 58.2031034330641
x28 = 77.0526593546029
x29 = 41.9330703621095
x30 = 10.5171438262116
x31 = 89.619029968962
x32 = 95.9022152761416
x33 = 16.8003291333912
x34 = -33.4651533240455
x35 = -23.4783055602705
x36 = -17.1951202530909
x37 = 51.9199181258845
x38 = 64.4862887402437
x39 = 70.7694740474233
x40 = 23.0835144405708
x41 = -29.7614908674501
x42 = 45.6367328187049
x43 = -48.6110467889889
x44 = 48.2162556692891
x45 = 26.7871768971662
x46 = -61.177417403348
x47 = -67.4606027105276
x48 = -64.8810798599435
x49 = -86.3101586320664
x50 = -127220.281664104
x51 = 98.4817381267258
x52 = 79.632182205187
x53 = -14.6155974025068
x54 = 111.048108741085
x55 = -4.62874963873176
x56 = -96.2970063958414
x57 = -90.0138210886618
x58 = 4.233958519032
x59 = 54.4994409764687
x60 = -27.1819680168659
x61 = -73.7437880177072
x62 = -3843.65497232546
x63 = 83.3358446617825
x64 = 7.93762097562742
x65 = 73.3489968980074
x66 = 29.3666997477503
x67 = -52.3147092455843
x68 = -58.5978945527639
x68 = -58.5978945527639
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