Question

answer is b but how​

Answer 1

Answer:

because the imaginary root of pie occurs in clockwise so sin is negitive

negitive of sinpie/4 is -1/root two

Answer 2

Question:

If Omega is the imaginary cube root of unity, then

$\sin( (\omega {}^{13} + \omega {}^{20})\pi + \frac{\pi}{4} )$

Theory :

Cube roots of unity are 1,ω,ω² Where ω and ω² are non- real complex cube roots .

•Properties

1) ω³ = 1

2)1+ω+ω² =0

⇒ω+ω² =-1

Solution :

We to find the value of

$\sin(( \omega {}^{13} + \omega {}^{20} ) \pi + \frac{\pi}{4} )$

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First solve :

$\omega {}^{13} + \omega {}^{20}$

$= ( \omega {}^{3} ) {}^{4} \times \omega + (\omega {}^{3} ) {}^{6} \times \omega {}^{2}$

Put the value of ω³=1

$= \omega + \omega {}^{2}$

$= - 1$

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Now :

$\sin(( \omega {}^{13} + \omega {}^{20})\pi + \frac{\pi}{4} )$

$put \: the \: value \: of \: \omega {}^{13} + \omega {}^{20} = - 1$

$= \sin( - \pi + \frac{\pi}{4} )$

$= \sin( \frac{ - 3\pi}{4} )$

$= - \sin( \frac{ - 3\pi}{4} )$

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

correct option b)