Question

solve it: sinthita+sin5thita=sin3thita​

Answer 1

Step-by-step explanation:

$\sf sin \thita + sin5 \theta = sin3 \theta \\ \\ \sf \therefore 2sin \frac{ \theta + 5 \theta}{2}cos \frac{ \theta - 5 \theta}{2} = sin3 \theta \\ \\ \sf \therefore 2sin \frac{6\theta}{2}cos{4 \theta}{2} - sin3 \theta \\ \\ \sf \therefore 2sin3 \theta cos2 \theta - sin3 \theta = 0 \\ \\ \sf \therefore sin3 \theta ( 2cos2 \theta - 1 ) = 0 \\ \\ \sf \therefore sin3 \theta = 0 \\ \\ \sf \therefore 3 \theta = n \pi \\ \\ \sf \boxed{\bold{\red{\sf \therefore \theta = \frac { n\pi}{3} }}} \\ \\ \sf \therefore cos2 \theta = \frac{1}{2} \\ \\ \sf \therefore 2 \theta = 2n\pi + \frac{\pi}{3} \\ \\ \sf \boxed{\bold{\red{\sf \therefore \theta = n \pi + \frac{\pi}{6}}}}$

Answer 2

$\begin{gathered}\sf sin \thita + sin5 \theta = sin3 \theta \\ \\ \sf \therefore 2sin \frac{ \theta + 5 \theta}{2}cos \frac{ \theta - 5 \theta}{2} = sin3 \theta \\ \\ \sf \therefore 2sin \frac{6\theta}{2}cos{4 \theta}{2} - sin3 \theta \\ \\ \sf \therefore 2sin3 \theta cos2 \theta - sin3 \theta = 0 \\ \\ \sf \therefore sin3 \theta ( 2cos2 \theta - 1 ) = 0 \\ \\ \sf \therefore sin3 \theta = 0 \\ \\ \sf \therefore 3 \theta = n \pi \\ \\ \sf \boxed{\bold{\red{\sf \therefore \theta = \frac { n\pi}{3} }}} \\ \\ \sf \therefore cos2 \theta = \frac{1}{2} \\ \\ \sf \therefore 2 \theta = 2n\pi + \frac{\pi}{3} \\ \\ \sf \boxed{\bold{\red{\sf \therefore \theta = n \pi + \frac{\pi}{6}}}}\end{gathered}$