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I have the following problem: I need to solve the equation: $$\frac{\cos\left(x\right)}{x^6} - \frac{6 \sin\left(x\right)}{x^7}==0$$

and I don't really remember how to do it symbolically.

Some help would be appreciated, thanks in advance.

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    $\begingroup$ Not an equation yet, are u saying the LHS=0? $\endgroup$ – thils Oct 10 '15 at 5:36
  • $\begingroup$ Is this Question about Mathematica? Else Math suits your needs better. $\endgroup$ – user9660 Oct 10 '15 at 5:38
  • $\begingroup$ I am in both the StockExchanges and I made a mistake - actually i meant to post it in Math. However, since I use Mathematica as well, I think it'd useful to learn also how to solve such equations symbolically with Mathematica... $\endgroup$ – user138364 Oct 10 '15 at 5:40
  • $\begingroup$ A transcendental equation like that is very unlikely to have a nice closed-form solution. $\endgroup$ – J. M. is away Oct 10 '15 at 5:45
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    $\begingroup$ The stock exchanges are interested in transcendental trig-poly equations? Should I be concerned about my 401K accounts? $\endgroup$ – Daniel Lichtblau Oct 11 '15 at 16:38
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Try this possibility

NSolve[Cos[x]/x^6 - 6 Sin[x]/x^7 == 0 && 10 < x < 30, x]

{{x -> 10.4754}, {x -> 13.725}, {x -> 16.9383}, {x -> 20.1307}, {x -> 
   23.31}, {x -> 26.4807}, {x -> 29.6454}}

All the x values indicates that x = 6 Tan[x] which can be deduced from your original equation.

If you use

Solve[Cos[x]/x^6 - 6 Sin[x]/x^7 == 0 && 10 < x < 30, x, Reals]

instead, you get

{{x -> Root[{-6 Sin[#1] + Cos[#1] #1 &, 
     10.4754178939933717338}]}, 
  {x -> 
   Root[{-6 Sin[#1] + Cos[#1] #1 &, 13.7250443836508641500}]},
  {x -> 
   Root[{-6 Sin[#1] + Cos[#1] #1 &, 16.9383246741988750253}]}, 
  {x -> 
   Root[{-6 Sin[#1] + Cos[#1] #1 &, 20.1306831271422053797}]}, 
  {x -> 
   Root[{-6 Sin[#1] + Cos[#1] #1 &, 23.3100136506053351414}]}, 
  {x -> 
   Root[{-6 Sin[#1] + Cos[#1] #1 &, 26.4807198099258826794}]},
  {x -> 
   Root[{-6 Sin[#1] + Cos[#1] #1 &, 29.6454356789935628843}]}}

And for all these x values, the root occurs at 6 Tan[x]

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    $\begingroup$ Nice answer. Suggest that you start at 3 rather than 10. First root around 3.7. $\endgroup$ – Jack LaVigne Oct 10 '15 at 14:30

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