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)

$\frac{1}{\sqrt{2}}$

Answer 2

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