Question

Which of the following statements is (are) CORRECT?
($Q$ denotes the set of rational numbers)

• A. If $\cos 2\theta \in Q$ and $\sin 2\theta \in Q,$ then $\tan \theta \in Q$ (if defined)
• B. If $\tan \theta \in Q,$ then $\sin 2\theta ,\cos 2\theta$ and $\tan 2\theta \in Q$ (if defined)
• C. If $\sin \theta \in Q$ and $\cos \theta \in Q,$ then $\tan 3\theta \in Q$ (if defined)
• D. If $\sin \theta \in Q,$ then $\cos 3\theta \in Q$

Answer 1

Answer: (A), (B), (C)

If $\sin \theta \in Q$ and $\cos \theta \in Q,$ then $\tan 3\theta \in Q$ (if defined)

$\tan \theta =\frac{1-\cos 2\theta }{\sin 2\theta }\in Q$
As $\cos 2\theta \in Q$ and $\sin 2\theta \in Q,$
$\therefore \tan \theta \in Q$

$\sin 2\theta =\frac{2\tan \theta }{1+{\tan }^2\theta }$
$\cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }$
$\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$
If $\tan \theta \in Q$, then we can clearly conclude from the above formulae that $\sin 2\theta ,\cos 2\theta$ and $\tan 2\theta \in Q$

If $\sin \theta ,\cos \theta \in Q,$
then $\sin 3\theta =3\sin \theta -4{\sin }^3\theta \in Q$
and $\cos 3\theta =4{\cos }^3\theta -3\cos \theta \in Q$
$\therefore \tan 3\theta =\frac{\sin 3\theta }{\cos 3\theta }\in Q$

If $\sin \theta \in Q,$
then $\sin 3\theta =3\sin \theta -4{\sin }^3\theta \in Q$
But $\cos 3\theta =\pm \sqrt{1-{\sin }^{2}3\theta }$ need not to be rational.
For e.g. $\sin \theta =\frac{1}{3}\in Q$
$\sin 3\theta =3\times \frac{1}{3}-4\times {\left(\frac{1}{3}\right)}^3=\frac{23}{27}$
But, $\cos 3\theta =\pm \frac{\sqrt{200}}{27}\notin Q$