Question

If \(ω\) is a complex cube root of unity, then the value of \(sin[π(ω^{10}+ω^{23})-\frac{π}{4}] =\)

• A. \(-\frac{\sqrt{3}}{2}\)
• B. \(-\frac{1}{\sqrt{2}}\)
• C. \(\frac{1}{\sqrt{2}}\)
• D. \(\frac{\sqrt{3}}{2}\)

Hint:

Reduce powers of omega using omega cubed = 1, then use 1 + w + w^2 = 0.

Answer 1

The Correct Option is C

Step 1: Powers of w
$23 = 3 \cdot 7 + 2$ and $10 = 3 \cdot 3 + 1$, so $\omega^{23} = \omega^2$ and $\omega^{10} = \omega$.

Step 2: Sum
$\omega + \omega^2 = -1$.

Step 3: Angle
The angle is $-\pi - \pi/4$. $\sin(-\pi - \theta) = \sin\theta$, so the value is $\sin(\pi/4) = 1/\sqrt{2}$. Option (C).

Final Answer:
Option (C). \[ \boxed{\frac{1}{\sqrt{2}}} \]