Question

If $\cos \theta =\sqrt{q}$ ,where $q$ is rational number and $\sqrt{q}$ is an irrational number, then find the integral values of $n$,for which $\cos n\theta$ is a rational number.

Answer 1

$cos\theta =\sqrt{q}$
Then $-1\le \sqrt{q}\le 1$
Now
$cos2\theta =2co{s}^2\theta -1$
$=2q-1$
Now $q$ is rational.
Hence
$2q^2-1$ is also rational.
Similarly
$cos4\theta =2co{s}^{2}2\theta -1$
$=2(2q-1{)}^2-1$
$=2(4q^2-4q+1)-1$
$=8q^2-8q+1$
Hence $cos4\theta$ is also rational.
And so on.
However $cos3\theta$
$=4co{s}^3\theta -3cos\theta$
$=\sqrt{q}[4q-3]$
Hence $cos3\theta$ is not rational.
Similarly $cos5\theta ,cos7\theta ,cos9\theta ..$ will not be rational.
Therefore we can conclude that $cosn\theta$ is rational is $nϵ\{2,4,\mathrm{6...2}N\}$, and irrational if $nϵ\{1,3,\mathrm{5..2}N+1\}$.