The number of solutions of the equation \cos 2 heta \cos \frac{ heta}{2} + \cos \frac{5 heta}{2} = 2\cos^3 \frac{5 heta}{2} in \left[ -\frac{\pi}{2}, \frac{\pi}{2} ight] is :

Question

The number of solutions of the equation \cos 2	heta \cos \frac{	heta}{2} + \cos \frac{5	heta}{2} = 2\cos^3 \frac{5	heta}{2} in \left[ -\frac{\pi}{2}, \frac{\pi}{2} ight] is :

Accepted Answer

Answer: B. 7 \begin{aligned} &\begin{aligned} & \cos 2 	heta \cos \frac{	heta}{2}+\cos \frac{5 	heta}{2}=2 \cos ^3 \frac{5 	heta}{2} \ & \frac{1}{2}\left(2 \cos 2 	heta \cos \frac{	heta}{2}ight)+\cos \frac{50}{2} \ & =\frac{1}{2}\left(\cos \frac{15 	heta}{2}+3 \cos \frac{5 	heta}{2}ight) \end{aligned}\ &	ext { or solving }\ &\begin{aligned} & \cos \frac{3 	heta}{2}=\cos \frac{15 	heta}{2} \ & \cos \frac{15 	heta}{2}-\cos \frac{3 	heta}{2}=0 \ & 2 \sin 30 \sin \frac{9 	heta}{2}=0 \end{aligned}\ &3 	heta=\mathrm{n} \pi \quad 	ext { or } \frac{9 	heta}{2}=\mathrm{m} \pi\ &\begin{aligned} & 	heta=\frac{\mathrm{n} \pi}{3} \quad 	heta=\frac{2 \mathrm{~m} \pi}{9} \ & 	heta=\left\{-\frac{\pi}{2}, \frac{\pi}{3}, 0ight\} \ & 	heta=\left\{-\frac{4 \pi}{9}, \frac{-2 \pi}{9}, \frac{4 \pi}{9}, \frac{2 \pi}{9}ight\} \end{aligned} \end{aligned}

The number of solutions of the equation $ \cos 2\theta \cos \frac{\theta}{2} + \cos \frac{5\theta}{2} = 2\cos^3 \frac{5\theta}{2} $ in $ \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] $ is :

Answer

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