1
$\begingroup$

Let $n$ be a positive integer. Can we precisely solve the equation $$\sin(x) = n\cos(x)$$ in $x$?

For $n=1$, we get $x=\pi/4$.

  • 1
    To add to the answer by @Fredrik, note that you don't lose any solutions by dividing by $\cos$ since both sides cannot be zero simultaneously.2011-12-01
  • 0
    The set of solutions is $x_n+\pi\mathbb Z$, for some $x_n$ in $[\pi/4,\pi/2)$ depending on $n\geqslant1$, such that $x_1=\pi/4$, $x_n\lt x_{n+1}$ for every $n$ and $x_n=\pi/2-1/n+o(1/n)$ when $n\to\infty$.2011-12-01
  • 1
    By the way: 18 minutes.2011-12-01

1 Answers 1

1

The only rational values of $\tan(x)$ or $\cot(x)$ for rational $x$ in degrees are $0$ and $\pm 1$. See for instance

Olmsted, J. M. H.; Discussions and Notes: Rational Values of Trigonometric Functions. Amer. Math. Monthly 52 (1945), no. 9, 507–508.

From this you cannot expect that your equation has any other nice solutions.

  • 1
    Related to this result is [Niven's theorem](http://mathworld.wolfram.com/NivensTheorem.html).2011-12-01