The sum of all values of [0,2 ] satisfying 2 ^2 = 2 and 2 ^2 =3 is

Question

The sum of all values of $	heta \in[0,2 \pi]$ satisfying $2 \sin ^2 	heta=\cos 2 	heta$ and $2 \cos ^2 	heta=3 \sin 	heta$ is

Accepted Answer

$$\begin{aligned}
& 2 \sin ^2 \theta=\cos 2 \theta \
& 2 \sin ^2 \theta=1-2 \sin ^2 \theta \
& 4 \sin ^2 \theta=1 \
& \sin ^2 \theta=\frac{1}{4} \
& \sin \theta= \pm \frac{1}{2} \
& 2 \cos ^2 \theta=3 \sin \theta \
& 2-2 \sin ^2 \theta+3 \sin \theta-2=0 \
& (2 \sin \theta-1)(2 \sin \theta-2)=0 \
& \sin \theta=\frac{1}{2}
\end{aligned}$$

so common equation which satisfy both equations is $\sin \theta=\frac{1}{2}$

$$\theta=\frac{\pi}{6}, \frac{5 \pi}{6} \quad(\theta \in[0,2 \pi])$$

$$\text { Sum }=\pi$$

Solutions · 1