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  4. If sin θ+ sin 2 θ+ sin 3 θ= sin α and cos θ+ cos 2 θ+ cos 3 θ= cos α, then θ is equal to:

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Q. If $\sin \theta+\sin 2 \theta+\sin 3 \theta=\sin \alpha$ and $\cos \theta+\cos 2 \theta+\cos 3 \theta=\cos \alpha$, then $\theta$ is equal to:

Trigonometric Functions

Solution:

$ \sin \theta+\sin 3 \theta+\sin 2 \theta=\sin \alpha$
$\Rightarrow 2 \sin 2 \theta \cos \theta+\sin 2 \theta=\sin \alpha$
$\Rightarrow \sin 2 \theta(2 \cos \theta+1)=\sin \alpha $ ... (i)
Now, $ \cos \theta+\cos 3 \theta+\cos 2 \theta=\cos \alpha$
$ \cos 2 \theta(2 \cos \theta+1)=\cos \alpha $... (ii)
From (i) and (ii),
$ \tan 2 \theta=\tan \alpha \Rightarrow 2 \theta=\alpha $
$ \Rightarrow \theta=\frac{\alpha}{2}$