The number of elements in the set S = [-4,4] : 3^22+62-10^2+5=0 is: Answer: 32

Question

The number of elements in the set $S = \left\{	heta\in[-4\pi,4\pi] : 3\cos^22	heta+6\cos2	heta-10\cos^2	heta+5=0ight\}$ is:

Answer: 32

Accepted Answer

Using $\cos^2\theta = \dfrac{1+\cos2\theta}2$:

$$3\cos^22\theta+6\cos2\theta-5(1+\cos2\theta)+5=0 \implies 3\cos^22\theta+\cos2\theta=0 \implies \cos2\theta(3\cos2\theta+1)=0$$

Case $\cos2\theta=0$: $2\theta=(2n+1)\dfrac\pi2$, giving $\theta = \pm\dfrac\pi4,\pm\dfrac{3\pi}4,\dots,\pm\dfrac{15\pi}4$ within $[-4\pi,4\pi]$ --- 16 solutions.

Case $\cos2\theta=-\dfrac13$: by the same density argument over $[-4\pi,4\pi]$ (8 full periods of $2\theta$), this also yields 16 solutions.

Total: $16+16 = 32$.

Solutions · 1