Question

The value of $\sin \left[n \pi+(-1)^{n} \frac{\pi}{4}\right], n \in I$ is

• A. $0$
• B. $\frac{1}{\sqrt{2}}$
• C. $-\frac{1}{\sqrt{2}}$
• D. None of these

Answer 1

Answer: (B)

$\sin \left[n \pi+(-1)^{n} \frac{\pi}{4}\right]=(-1)^{n} \sin \left[(-1)^{n} \frac{\pi}{4}\right]$
$\left[\because \sin (n \pi+\theta)=(-1)^{n} \sin \theta\right]$
$=(-1)^{n}(-1)^{n} \sin \frac{\pi}{4}$
$\left.\therefore \sin \left[(-1)^{n} \theta\right]=(-1)^{n} \sin \theta\right]=(-1)^{2 n} \sin \frac{\pi}{4}$
$=\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}$