Question
If $$\omega$$ is imaginary cube root of unity, then $$\sin \left\{ {\left( {{\omega ^{13}} + {\omega ^2}} \right)\pi + \frac{\pi }{4}} \right\}$$ is equal to
A.
$$ - \frac{{\sqrt 3 }}{2}$$
B.
$$ - \frac{1}{{\sqrt 2 }}$$
C.
$$ \frac{1}{{\sqrt 2 }}$$
D.
$$\frac{{\sqrt 3 }}{2}$$
Answer :
$$ - \frac{1}{{\sqrt 2 }}$$
Solution :
$$\eqalign{
& \sin \left\{ {\left( {{\omega ^{13}} + {\omega ^2}} \right)\pi + \frac{\pi }{4}} \right\} \cr
& = \sin \left\{ {\left( {\omega + {\omega ^2}} \right)\pi + \frac{\pi }{4}} \right\} = \sin \left( { - \pi + \frac{\pi }{4}} \right) \cr
& = - \sin \frac{\pi }{4} = - \frac{1}{{\sqrt 2 }} \cr} $$