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Q)

# Evaluate :$\sin(n\pi+(-1)^n\large\frac{\pi}{4})$ where $n$ is an integer,

$\begin{array}{1 1}(A)\;\large\frac{1}{\sqrt 2}&(B)\;\large\frac{1}{2}\\(C)\;1&(D)\;0\end{array}$

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A)
Toolbox:
• $\sin (\pi+\theta)=-\sin \theta$
• $\sin(n\pi+\theta)=(-1)^n\sin \theta$
• $\sin(-\theta)=-\sin \theta$
$\sin(n\pi+(-1)^n\large\frac{\pi}{4})$
$\Rightarrow (-1)^n\sin[(-1)^n\large\frac{\pi}{4}]$
$\sin(n\pi+\theta)=(-1)^n\sin \theta$
$\Rightarrow (-1)^n(-1)^n\sin \large\frac{\pi}{4}$
$\sin(-\theta)=-\sin \theta$
$\Rightarrow (-1)^{2n}\sin \large\frac{\pi}{4}$
$\Rightarrow \sin \large\frac{\pi}{4}$
$2n$ is even
$\Rightarrow \large\frac{1}{\sqrt 2}$
Hence (A) is the correct answer.