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sin3x=cos5x la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
sin(3*x) = cos(5*x)
$$\sin{\left(3 x \right)} = \cos{\left(5 x \right)}$$
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Respuesta rápida [src] 
     -pi 
x1 = ----
      4
$$x_{1} = - \frac{\pi}{4}$$ 
     pi
x2 = --
     16
$$x_{2} = \frac{\pi}{16}$$ 
     3*pi
x3 = ----
      4
$$x_{3} = \frac{3 \pi}{4}$$ 
           / 16____\
x4 = -I*log\-\/ -1 /
$$x_{4} = - i \log{\left(- \sqrt[16]{-1} \right)}$$ 
              /   /pi\\                                   
              |cos|--||        /    _____________________\
              |   \16/|        |   /    2/pi\      2/pi\ |
x5 = pi - atan|-------| - I*log|  /  cos |--| + sin |--| |
              |   /pi\|        \\/       \16/       \16/ /
              |sin|--||                                   
              \   \16//
$$x_{5} = - \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{16} \right)}}{\sin{\left(\frac{\pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)} + \pi$$ 
           /   /pi\\                                   
           |cos|--||        /    _____________________\
           |   \16/|        |   /    2/pi\      2/pi\ |
x6 = - atan|-------| - I*log|  /  cos |--| + sin |--| |
           |   /pi\|        \\/       \16/       \16/ /
           |sin|--||                                   
           \   \16//
$$x_{6} = - \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{16} \right)}}{\sin{\left(\frac{\pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)}$$ 
                                                      /   /3*pi\\
                /    _________________________\       |cos|----||
                |   /    2/3*pi\      2/3*pi\ |       |   \ 16 /|
x7 = -pi - I*log|  /  cos |----| + sin |----| | + atan|---------|
                \\/       \ 16 /       \ 16 / /       |   /3*pi\|
                                                      |sin|----||
                                                      \   \ 16 //
$$x_{7} = - \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \cos^{2}{\left(\frac{3 \pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)}$$ 
                                                  /   /5*pi\\
            /    _________________________\       |sin|----||
            |   /    2/3*pi\      2/5*pi\ |       |   \ 16 /|
x8 = - I*log|  /  sin |----| + sin |----| | + atan|---------|
            \\/       \ 16 /       \ 16 / /       |   /3*pi\|
                                                  |sin|----||
                                                  \   \ 16 //
$$x_{8} = - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\sin{\left(\frac{5 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)}$$ 
              /   /3*pi\\                                       
              |sin|----||        /    _________________________\
              |   \ 16 /|        |   /    2/3*pi\      2/5*pi\ |
x9 = pi - atan|---------| - I*log|  /  sin |----| + sin |----| |
              |   /5*pi\|        \\/       \ 16 /       \ 16 / /
              |sin|----||                                       
              \   \ 16 //
$$x_{9} = - \operatorname{atan}{\left(\frac{\sin{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{5 \pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)} + \pi$$ 
            /   /3*pi\\                                       
            |sin|----||        /    _________________________\
            |   \ 16 /|        |   /    2/3*pi\      2/5*pi\ |
x10 = - atan|---------| - I*log|  /  sin |----| + sin |----| |
            |   /5*pi\|        \\/       \ 16 /       \ 16 / /
            |sin|----||                                       
            \   \ 16 //
$$x_{10} = - \operatorname{atan}{\left(\frac{\sin{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{5 \pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)}$$ 
x10 = -atan(sin(3*pi/16)/sin(5*pi/16)) - i*log(sqrt(sin(3*pi/16)^2 + sin(5*pi/16)^2))
Suma y producto de raíces [src] 
suma
                                             /   /pi\\                                            /   /pi\\                                                                                       /   /3*pi\\                                                /   /5*pi\\            /   /3*pi\\                                                /   /3*pi\\                                       
                                             |cos|--||        /    _____________________\         |cos|--||        /    _____________________\              /    _________________________\       |cos|----||          /    _________________________\       |sin|----||            |sin|----||        /    _________________________\         |sin|----||        /    _________________________\
  pi   pi   3*pi        / 16____\            |   \16/|        |   /    2/pi\      2/pi\ |         |   \16/|        |   /    2/pi\      2/pi\ |              |   /    2/3*pi\      2/3*pi\ |       |   \ 16 /|          |   /    2/3*pi\      2/5*pi\ |       |   \ 16 /|            |   \ 16 /|        |   /    2/3*pi\      2/5*pi\ |         |   \ 16 /|        |   /    2/3*pi\      2/5*pi\ |
- -- + -- + ---- - I*log\-\/ -1 / + pi - atan|-------| - I*log|  /  cos |--| + sin |--| | + - atan|-------| - I*log|  /  cos |--| + sin |--| | + -pi - I*log|  /  cos |----| + sin |----| | + atan|---------| + - I*log|  /  sin |----| + sin |----| | + atan|---------| + pi - atan|---------| - I*log|  /  sin |----| + sin |----| | + - atan|---------| - I*log|  /  sin |----| + sin |----| |
  4    16    4                               |   /pi\|        \\/       \16/       \16/ /         |   /pi\|        \\/       \16/       \16/ /              \\/       \ 16 /       \ 16 / /       |   /3*pi\|          \\/       \ 16 /       \ 16 / /       |   /3*pi\|            |   /5*pi\|        \\/       \ 16 /       \ 16 / /         |   /5*pi\|        \\/       \ 16 /       \ 16 / /
                                             |sin|--||                                            |sin|--||                                                                                       |sin|----||                                                |sin|----||            |sin|----||                                                |sin|----||                                       
                                             \   \16//                                            \   \16//                                                                                       \   \ 16 //                                                \   \ 16 //            \   \ 16 //                                                \   \ 16 //
$$\left(- \operatorname{atan}{\left(\frac{\sin{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{5 \pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)}\right) + \left(\left(\left(\left(- \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \cos^{2}{\left(\frac{3 \pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)}\right) + \left(\left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{16} \right)}}{\sin{\left(\frac{\pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)}\right) + \left(\left(- i \log{\left(- \sqrt[16]{-1} \right)} + \left(\left(- \frac{\pi}{4} + \frac{\pi}{16}\right) + \frac{3 \pi}{4}\right)\right) + \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{16} \right)}}{\sin{\left(\frac{\pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)} + \pi\right)\right)\right)\right) + \left(- i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\sin{\left(\frac{5 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)}\right)\right) + \left(- \operatorname{atan}{\left(\frac{\sin{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{5 \pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)} + \pi\right)\right)$$ 
=
        /   /pi\\         /   /3*pi\\                                                                                                                                                     /   /3*pi\\       /   /5*pi\\
        |cos|--||         |sin|----||                /    _________________________\                           /    _________________________\          /    _____________________\       |cos|----||       |sin|----||
        |   \16/|         |   \ 16 /|   25*pi        |   /    2/3*pi\      2/3*pi\ |        / 16____\          |   /    2/3*pi\      2/5*pi\ |          |   /    2/pi\      2/pi\ |       |   \ 16 /|       |   \ 16 /|
- 2*atan|-------| - 2*atan|---------| + ----- - I*log|  /  cos |----| + sin |----| | - I*log\-\/ -1 / - 3*I*log|  /  sin |----| + sin |----| | - 2*I*log|  /  cos |--| + sin |--| | + atan|---------| + atan|---------|
        |   /pi\|         |   /5*pi\|     16         \\/       \ 16 /       \ 16 / /                           \\/       \ 16 /       \ 16 / /          \\/       \16/       \16/ /       |   /3*pi\|       |   /3*pi\|
        |sin|--||         |sin|----||                                                                                                                                                     |sin|----||       |sin|----||
        \   \16//         \   \ 16 //                                                                                                                                                     \   \ 16 //       \   \ 16 //
$$- i \log{\left(- \sqrt[16]{-1} \right)} - 2 \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{16} \right)}}{\sin{\left(\frac{\pi}{16} \right)}} \right)} - 2 \operatorname{atan}{\left(\frac{\sin{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{5 \pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \cos^{2}{\left(\frac{3 \pi}{16} \right)}} \right)} - 3 i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)} - 2 i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\sin{\left(\frac{5 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)} + \frac{25 \pi}{16}$$ 
producto
                               /         /   /pi\\                                   \ /      /   /pi\\                                   \ /                                                 /   /3*pi\\\ /                                             /   /5*pi\\\ /         /   /3*pi\\                                       \ /      /   /3*pi\\                                       \
                               |         |cos|--||        /    _____________________\| |      |cos|--||        /    _____________________\| |           /    _________________________\       |cos|----||| |       /    _________________________\       |sin|----||| |         |sin|----||        /    _________________________\| |      |sin|----||        /    _________________________\|
-pi  pi 3*pi /      / 16____\\ |         |   \16/|        |   /    2/pi\      2/pi\ || |      |   \16/|        |   /    2/pi\      2/pi\ || |           |   /    2/3*pi\      2/3*pi\ |       |   \ 16 /|| |       |   /    2/3*pi\      2/5*pi\ |       |   \ 16 /|| |         |   \ 16 /|        |   /    2/3*pi\      2/5*pi\ || |      |   \ 16 /|        |   /    2/3*pi\      2/5*pi\ ||
----*--*----*\-I*log\-\/ -1 //*|pi - atan|-------| - I*log|  /  cos |--| + sin |--| ||*|- atan|-------| - I*log|  /  cos |--| + sin |--| ||*|-pi - I*log|  /  cos |----| + sin |----| | + atan|---------||*|- I*log|  /  sin |----| + sin |----| | + atan|---------||*|pi - atan|---------| - I*log|  /  sin |----| + sin |----| ||*|- atan|---------| - I*log|  /  sin |----| + sin |----| ||
 4   16  4                     |         |   /pi\|        \\/       \16/       \16/ /| |      |   /pi\|        \\/       \16/       \16/ /| |           \\/       \ 16 /       \ 16 / /       |   /3*pi\|| |       \\/       \ 16 /       \ 16 / /       |   /3*pi\|| |         |   /5*pi\|        \\/       \ 16 /       \ 16 / /| |      |   /5*pi\|        \\/       \ 16 /       \ 16 / /|
                               |         |sin|--||                                   | |      |sin|--||                                   | |                                                 |sin|----||| |                                             |sin|----||| |         |sin|----||                                       | |      |sin|----||                                       |
                               \         \   \16//                                   / \      \   \16//                                   / \                                                 \   \ 16 /// \                                             \   \ 16 /// \         \   \ 16 //                                       / \      \   \ 16 //                                       /
$$- i \log{\left(- \sqrt[16]{-1} \right)} \frac{3 \pi}{4} \cdot - \frac{\pi}{4} \frac{\pi}{16} \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{16} \right)}}{\sin{\left(\frac{\pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)} + \pi\right) \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\pi}{16} \right)}}{\sin{\left(\frac{\pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)}\right) \left(- \pi - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \cos^{2}{\left(\frac{3 \pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)}\right) \left(- i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)} + \operatorname{atan}{\left(\frac{\sin{\left(\frac{5 \pi}{16} \right)}}{\sin{\left(\frac{3 \pi}{16} \right)}} \right)}\right) \left(- \operatorname{atan}{\left(\frac{\sin{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{5 \pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)} + \pi\right) \left(- \operatorname{atan}{\left(\frac{\sin{\left(\frac{3 \pi}{16} \right)}}{\sin{\left(\frac{5 \pi}{16} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{3 \pi}{16} \right)} + \sin^{2}{\left(\frac{5 \pi}{16} \right)}} \right)}\right)$$ 
=
        /         /      ___________  \\ /         /      ___________  \\ /          /       ___________\\     /      ___________  \     /      ___________  \     /       ___________\             
        |         |     /       ___   || |         |     /       ___   || |          |      /       ___ ||     |     /       ___   |     |     /       ___   |     |      /       ___ |             
      3 |         |   \/  2 - \/ 2    || |         |   \/  2 + \/ 2    || |          |2 - \/  2 - \/ 2  ||     |   \/  2 - \/ 2    |     |   \/  2 + \/ 2    |     |2 - \/  2 - \/ 2  |    / 16____\
3*I*pi *|pi + atan|-------------------||*|pi + atan|-------------------||*|-pi + atan|------------------||*atan|-------------------|*atan|-------------------|*atan|------------------|*log\-\/ -1 /
        |         |        ___________|| |         |        ___________|| |          |     ___________  ||     |        ___________|     |        ___________|     |     ___________  |             
        |         |       /       ___ || |         |       /       ___ || |          |    /       ___   ||     |       /       ___ |     |       /       ___ |     |    /       ___   |             
        \         \-2 + \/  2 + \/ 2  // \         \-2 + \/  2 - \/ 2  // \          \  \/  2 + \/ 2    //     \-2 + \/  2 + \/ 2  /     \-2 + \/  2 - \/ 2  /     \  \/  2 + \/ 2    /             
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                256
$$\frac{3 i \pi^{3} \left(- \pi + \operatorname{atan}{\left(\frac{2 - \sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}\right) \left(\operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{-2 + \sqrt{2 - \sqrt{2}}} \right)} + \pi\right) \left(\operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{-2 + \sqrt{\sqrt{2} + 2}} \right)} + \pi\right) \log{\left(- \sqrt[16]{-1} \right)} \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{-2 + \sqrt{2 - \sqrt{2}}} \right)} \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{-2 + \sqrt{\sqrt{2} + 2}} \right)} \operatorname{atan}{\left(\frac{2 - \sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}}{256}$$ 
3*i*pi^3*(pi + atan(sqrt(2 - sqrt(2))/(-2 + sqrt(2 + sqrt(2)))))*(pi + atan(sqrt(2 + sqrt(2))/(-2 + sqrt(2 - sqrt(2)))))*(-pi + atan((2 - sqrt(2 - sqrt(2)))/sqrt(2 + sqrt(2))))*atan(sqrt(2 - sqrt(2))/(-2 + sqrt(2 + sqrt(2))))*atan(sqrt(2 + sqrt(2))/(-2 + sqrt(2 - sqrt(2))))*atan((2 - sqrt(2 - sqrt(2)))/sqrt(2 + sqrt(2)))*log(-(-1)^(1/16))/256
Respuesta numérica [src] 
x1 = 4.1233403578366
x2 = -13.9408174003047
x3 = -69.7040870015235
x4 = -98.7638190472291
x5 = 10.4065256650162
x6 = -17.0824100538945
x7 = 32.3976742401447
x8 = 37.8954613839269
x9 = -95.6222263936393
x10 = -6.08683576633022
x11 = 5.49778714378214
x12 = 84.2339530243763
x13 = 96.0149254753381
x14 = 44.1786466911065
x15 = -97.9784208838317
x16 = 14.3335164820034
x17 = 12.7627201552085
x18 = 70.0967860832223
x19 = -91.6952355766521
x20 = -7.65763209312512
x21 = -72.0602814917159
x22 = 68.329640215578
x23 = -47.712938426395
x24 = 81.0923603707865
x25 = -85.6083998103219
x26 = 48.1056375080937
x27 = -21.7947990342792
x28 = -65.7770961845363
x29 = 11.9773219918111
x30 = 90.3207887907066
x31 = -87.7682447596649
x32 = -82.4668071567321
x33 = -75.9872723087031
x34 = -41.6261026600648
x35 = 26.1144889329652
x36 = 54.3888228152733
x37 = 27.6852852597601
x38 = -68.1332906747286
x39 = 15.9043128087983
x40 = 62.2428044492478
x41 = -19.6349540849362
x42 = -57.9231145505618
x43 = 24.3473430653209
x44 = -73.6310778185108
x45 = -94.0514300668444
x46 = -50.0691329165873
x47 = -10.0138265833175
x48 = 81.877758534184
x49 = 92.0879346583508
x50 = -79.9142631256904
x51 = -29.6487806682537
x52 = -17.8678082172919
x53 = -53.9961237335746
x54 = -43.7859476094077
x55 = -61.8501053675491
x56 = 52.0326283250809
x57 = 8.05033117482385
x58 = 63.0282026126452
x59 = 99.9419162923253
x60 = -25.7217898512664
x61 = 33.9684705669396
x62 = -90.1244392498572
x63 = -11.5846229101124
x64 = -83.8412539426776
x65 = 77.9507677171967
x66 = 66.169795266235
x67 = -46.1421420996001
x68 = -3.73064127613788
x69 = -33.5757714852409
x70 = 36.324665057132
x71 = 76.3799713904018
x72 = -69.9004365423729
x73 = -24.1509935244715
x74 = 0.196349540849362
x75 = 728.064097469435
x76 = -61.0647072041516
x77 = 46.3384916404494
x78 = -51.6399292433822
x79 = -63.6172512351933
x80 = 55.9596191420682
x81 = -32.004975158446
x82 = 2.35619449019234
x83 = 58.3158136322605
x84 = -79.1288649622929
x85 = 40.2516558741192
x86 = 80.3069622073891
x87 = 30.0414797499524
x88 = -28.0779843414588
x89 = 98.3711199655304
x90 = 18.2605072989907
x91 = 59.8866099590554
x92 = -2.15984494934298
x93 = -35.9319659754333
x94 = 22.1874981159779
x95 = 74.0237769002095
x96 = -39.8589567924205
x97 = -55.5669200603695
x98 = 88.1609438413636
x99 = 19.8313036257856
x99 = 19.8313036257856
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