Question

The number of solutions of equation $6\, cos\,2\theta + 2\, cos^2(\theta/2) + 2\, sin^2 \theta = 0, -\pi < \theta < \pi$ is

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (A)

$ 6 \,cos\,2\theta + 2 \,cos^2(\theta/2) + 2 \,sin^2\theta = 0$
or $12 \,cos^2\theta - 6 + 1 + cos\,\theta + 2 - 2 \,cos^2\theta = 0$
or $10 \,cos^2 \theta + cos\,\theta - 3 = 0$
or $(5\, cos\,\theta + 3) (2\, cos\,\theta - 1) = 0$
or $cos\,\theta = -\frac{3}{5}, \frac{1}{2}$
$\Rightarrow \theta = \frac{\pi}{3}, \pi - cos^{-1} (\frac{3}{5}), - \frac{\pi}{3}, - \pi + cos^{-1} (\frac{3}{5})$