The sum of all values of $ heta\in[0,2\pi]$ satisfying $2\sin^2 heta=\cos2 heta$ and $2\cos^2 heta=3\sin heta$ is

Question

The sum of all values of $	heta\in[0,2\pi]$ satisfying $2\sin^2	heta=\cos2	heta$ and $2\cos^2	heta=3\sin	heta$ is

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] The second equation gives $2s^2+3s-2=0$, so $\sin	heta=\frac12$ (since $\sin	heta=-2$ is impossible). The first equation then reads $2\cdot\frac14=1-2\cdot\frac14$, which holds. So $	heta=\frac\pi6,\frac{5\pi}6$ and the sum is $\pi$. Answer (3).

Q. The sum of all values of $\theta\in[0,2\pi]$ satisfying $2\sin^2\theta=\cos2\theta$ and $2\cos^2\theta=3\sin\theta$ is

Explanation

Reviewed by Dr. Abubacker

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