First slide

Multiple and sub- multiple Angles

Question

Which of the following statements are always correct (where Q denotes the set of ratationals)?

Moderate

A

$\cos 2 θ \in Q and \sin 2 θ \in Q \Rightarrow \tan θ \in Q (if defined)$
B

$\tan θ \in Q \Rightarrow \sin 2 θ, \cos 2 θ and \tan 2 θ \in Q (if defined)$
C

$if \sin θ \in Q and \cos θ \in Q \Rightarrow \tan 3 θ \in Q (if defined)$
D

$if \sin θ \in Q \Rightarrow \cos 3 θ \in Q$

Solution

(1) $\tan θ = \frac{1 - \cos 2 θ}{\sin 2 θ}$ . Hence, option (l) is correct.
$\begin{matrix} (2) \sin 2 θ = \frac{2 \tan θ}{1 + \tan^{2} θ} \\ \cos 2 θ = \frac{1 - \tan^{2} θ}{1 + \tan^{2} θ} \\ \tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ} \end{matrix}$
Hence, option (2) is correct.
(3) $\tan 3 θ = \frac{\sin 3 θ}{\cos 3 θ}$ . Hence, option (3) is correct.
(4) $\sin θ = 1 / 3$ , which is rational but $\cos 3 θ = \cos θ (4 \cos^{2} θ - 3)$ ,
which is irrational.
Hence, option (4) is incorrect.

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