Q. If $n$ is any integer, then the general solution of the equation $\cos{\theta - \sin{\theta = \frac{1}{\sqrt{2}}}}$ is
A) $\theta = 2n\pi - \frac{\pi}{12}$ or $\theta = 2n\pi + \frac{7\pi}{12}$
B) $\theta = n\pi + \frac{\pi}{12}$
C) $\theta = 2n\pi + \frac{\pi}{12}$ or $\theta = 2n\pi - \frac{7\pi}{12}$
D) $\theta = 2n\pi + \frac{\pi}{12}$ or $\theta = 2n\pi + \frac{7\pi}{12}$
Explanation
(c)
Given, $\cos{\theta - \sin{\theta = \frac{1}{\sqrt{2}}}}$
$\Rightarrow \ \frac{1}{\sqrt{2}}\cos{\theta - \frac{1}{\sqrt{2}}\sin{\theta = \frac{1}{2}}}$
$\Rightarrow \ \cos\left( \theta + \frac{\pi}{4} \right) = \cos\frac{\pi}{3}$
$\Rightarrow \ \ \theta + \frac{\pi}{4} = 2n\pi \pm \frac{\pi}{3}$
$\Rightarrow \ \ \ \ \theta = 2n\pi - \frac{7\pi}{12}$ or $2n\pi + \frac{\pi}{12}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.