The number of solutions of the equation 2 ()/(2) + (5)/(2) = 2^3 (5)/(2) in [ -()/(2), ()/…

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

$$\begin{aligned}
&\begin{aligned}
& \cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^3 \frac{5 \theta}{2} \
& \frac{1}{2}\left(2 \cos 2 \theta \cos \frac{\theta}{2}\right)+\cos \frac{50}{2} \
& =\frac{1}{2}\left(\cos \frac{15 \theta}{2}+3 \cos \frac{5 \theta}{2}\right)
\end{aligned}\
\&\text { or solving }\
\&\begin{aligned}
& \cos \frac{3 \theta}{2}=\cos \frac{15 \theta}{2} \
& \cos \frac{15 \theta}{2}-\cos \frac{3 \theta}{2}=0 \
& 2 \sin 30 \sin \frac{9 \theta}{2}=0
\end{aligned}\
\&3 \theta=\mathrm{n} \pi \quad \text { or } \frac{9 \theta}{2}=\mathrm{m} \pi\
\&\begin{aligned}
& \theta=\frac{\mathrm{n} \pi}{3} \quad \theta=\frac{2 \mathrm{~m} \pi}{9} \
& \theta=\left\{-\frac{\pi}{2}, \frac{\pi}{3}, 0\right\} \
& \theta=\left\{-\frac{4 \pi}{9}, \frac{-2 \pi}{9}, \frac{4 \pi}{9}, \frac{2 \pi}{9}\right\}
\end{aligned}
\end{aligned}$$

Solutions · 1