First slide

Multiple and sub- multiple Angles

Question

If ${(\frac{\sin θ}{\sin ϕ})}^{2} = \frac{\tan θ}{\tan ϕ} = 3 t h e n$ which of the following is not true.

Moderate

A

$\tan ϕ = \frac{1}{\sqrt{3}}$
B

$\tan ϕ = \frac{- 1}{\sqrt{3}}$
C

$\tan θ = \sqrt{3}$
D

$\tan θ = \frac{1}{\sqrt{3}}$

Solution

$\begin{array}{l} W e h a v e {(\frac{\sin θ}{\sin ϕ})}^{2} = \frac{\tan θ}{\tan ϕ} = 3 \\ \Rightarrow \frac{\sin θ}{\sin ϕ} . \frac{\sin θ}{\sin ϕ} = \frac{\sin θ}{\sin ϕ} . \frac{\cos ϕ}{\cos θ} \\ \Rightarrow \sin 2 θ = \sin 2 ϕ \\ \Rightarrow \frac{2 \tan θ}{1 + \tan^{2} θ} = \frac{2 \tan ϕ}{1 + \tan^{2} ϕ} \\ \Rightarrow \frac{6 \tan ϕ}{1 + 9 \tan^{2} ϕ} = \frac{2 \tan ϕ}{1 + \tan^{2} ϕ} (∵ \tan θ = 3 \tan ϕ) \\ \Rightarrow \tan^{2} ϕ = \frac{1}{3} \\ ⇒ \tan ϕ = \pm \frac{1}{\sqrt{3}} \\ ⇒ \tan θ = \pm \sqrt{3} \end{array}$

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