2
$\begingroup$

I was able to reduce an equation I have to:

$$f(t) = \tan(\mu) \tan(\nu) - C = 0$$

where $\mu, \nu$ are linear functions of t and $C$ is a constant.

  1. Are there any identities for the product of tangents?
  2. Is there a way to solve this equation analytically?
  • 0
    You should give explicitely $\mu$ and $\nu$ as functions of time. This can simplify a lot.2012-02-09
  • 1
    For 1., [sure, there are](http://functions.wolfram.com/ElementaryFunctions/Tan/16/05/01/0001/). I doubt that they'd be helpful here, though. For 2., as mentioned a number of times before, transcendental equations usually don't admit easy analytical solutions; what makes you think it would be different here?2012-02-09
  • 0
    Special case: $\mu=t$, $\nu=5t$. There's probably an identity for $\tan5t$ in terms of $(\tan t)^5$ which will lead to some polynomial equation of high degree in $\tan t$ and thereby to numerical methods of solution only.2012-02-09
  • 0
    @Jon: They're just run-of-the-mill linear functions. ie: http://en.wikipedia.org/wiki/Linear_function2012-02-09
  • 0
    @J.M.: If one of $\mu$ or $\nu$ are constants, or $C$ is $0$, it's solvable analytically. So there was enough reason for me to suspect an analytic solution might be possible.2012-02-09

3 Answers 3