Question

If \( \omega \) is a complex cube root of unity, then \[ \cos\left(\left(\omega^{1234} + \omega^{2021}\right)\pi - \frac{\pi}{4}\right) = \]

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

Hint:

For cube roots of unity: \(\omega^3 = 1\), \(1+\omega+\omega^2=0\), so \(\omega+\omega^2=-1\). Reduce exponents mod 3.

Answer 1

The Correct Option is A

Concept: For complex cube roots of unity, \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \), with \( \omega^2 = \overline{\omega} \). Powers of \( \omega \) repeat every 3: \[ \omega^{3k} = 1,\quad \omega^{3k+1} = \omega,\quad \omega^{3k+2} = \omega^2. \]

Step 1: Reduce powers modulo 3. \[ 1234 \div 3: \quad 3 \times 411 = 1233, \text{ remainder } 1 \quad \Rightarrow \quad \omega^{1234} = \omega^1 = \omega. \] \[ 2021 \div 3: \quad 3 \times 673 = 2019, \text{ remainder } 2 \quad \Rightarrow \quad \omega^{2021} = \omega^2. \]

Step 2: Sum the powers. \[ \omega^{1234} + \omega^{2021} = \omega + \omega^2. \] Since \( 1 + \omega + \omega^2 = 0 \), we have \( \omega + \omega^2 = -1 \).

Step 3: Substitute into the cosine argument. \[ \left(\omega^{1234} + \omega^{2021}\right)\pi - \frac{\pi}{4} = (-1)\pi - \frac{\pi}{4} = -\pi - \frac{\pi}{4} = -\frac{5\pi}{4}. \]

Step 4: Evaluate the cosine. \[ \cos\left(-\frac{5\pi}{4}\right) = \cos\left(\frac{5\pi}{4}\right) \quad (\text{since cosine is even}). \] \[ \frac{5\pi}{4} = \pi + \frac{\pi}{4}, \quad \cos\left(\pi + \frac{\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}. \]