First slide

Algebra of complex numbers

Question

If $ω$ is an imaginary cube root of unity, then the value of $\sin \{(ω^{10} + ω^{23}) π - \frac{π}{4}\}$ is

Moderate

A

$- \frac{\sqrt{3}}{2}$
B

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

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

$\frac{\sqrt{3}}{2}$

Solution

Using , $ω^{10} = ω, ω^{23} = ω^{2}$ , we get
$\sin \{(ω^{10} + ω^{23}) π - \frac{π}{4}\} = \sin (- π - \frac{π}{4}) = \sin \frac{π}{4} = \frac{1}{\sqrt{2}}$

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