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: Understanding the Concept
For a complex cube root of unity, \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\). So powers repeat every 3.

Step 2: Reduce the powers
\(\omega^{10} = \omega^{9}\cdot\omega = \omega\). \(\omega^{23} = \omega^{21}\cdot\omega^2 = \omega^2\).
\[ \omega^{10} + \omega^{23} = \omega + \omega^2 = -1 \]

Step 3: Evaluate the sine
\[ \sin\left[\pi(-1) - \frac{\pi}{4}\right] = \sin\left(-\frac{5\pi}{4}\right) = -\sin\frac{5\pi}{4} \]
Since \(\sin\frac{5\pi}{4} = -\frac{1}{\sqrt 2}\), the value is \(+\frac{1}{\sqrt 2}\). The negative options come from a sign slip, and (D) is not a sine of a multiple of \(\pi/4\).

Final Answer:
The value is \(\frac{1}{\sqrt 2}\), option (C). \[ \boxed{\frac{1}{\sqrt{2}}} \]