# How to solve trigonometric equations

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.

• Not an equation yet, are u saying the LHS=0? – thils Oct 10 '15 at 5:36
• Is this Question about Mathematica? Else Math suits your needs better. – user9660 Oct 10 '15 at 5:38
• 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... – user138364 Oct 10 '15 at 5:40
• A transcendental equation like that is very unlikely to have a nice closed-form solution. – J. M.'s ennui Oct 10 '15 at 5:45
• The stock exchanges are interested in transcendental trig-poly equations? Should I be concerned about my 401K accounts? – Daniel Lichtblau Oct 11 '15 at 16:38

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]


{{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]

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