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Saturday, March 30, 2024

2024/021) Show that \sin 3\theta = \sin \theta + 2 \sin \theta \cos 2\theta and use it to prove \sin \frac{\pi}{9} + 2 \sin \frac{\pi}{9} \cos \frac{2\pi}{9} = \frac{\sqrt{3}}{2}

LHS

= \sin 3 \theta = \sin ( \theta + 2\theta) = \ sin \theta \cos 2 \theta + \cos \theta \sin 2\theta usng \sin(A+B) formula 

= \ sin \theta \cos 2 \theta + \cos \theta (2\sin \theta \cos \theta) using \sin 2\theta formula

= \ sin \theta (\cos 2 \theta + 2  \cos^2  \theta)

= \ sin \theta ( 2 \cos 2 \theta +  \cos 2  \theta + 1 ) using formula for \cos 2\theta

= \ sin \theta ( 2 \cos 2 \theta +   1 )

= 2 \ sin \theta  \cos 2 \theta +  \sin   \theta which is RHS

In the above putting \theta = \frac{\pi}{3}  we get

\sin \frac{\pi}{9} +  2 \sin \frac{\pi}{9} \cos \frac{2\pi}{9} = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}

 

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