Question

The complex number \( \sqrt{2}\left[ \sin \frac{\pi}{8} + i \cos \frac{\pi}{8} \right]^6 \) represents

• A. \( -i \)
• B. \( i \)
• C. \( 1 - i \)
• D. \( 1 + i \)
• E. \( 1 + 2i \)

Hint:

Convert expressions into \( \cos\theta + i\sin\theta \) form before applying powers.

Answer 1

The Correct Option is A

Concept: Use De Moivre's theorem: \[ (\cos \theta + i\sin \theta)^n = \cos(n\theta) + i\sin(n\theta) \]

Step 1: Rewrite expression. \[ \sin \frac{\pi}{8} + i\cos \frac{\pi}{8} = \cos\left(\frac{\pi}{2} - \frac{\pi}{8}\right) + i\sin\left(\frac{\pi}{2} - \frac{\pi}{8}\right) \] \[ = \cos\frac{3\pi}{8} + i\sin\frac{3\pi}{8} \]

Step 2: Apply power. \[ \left[\cos\frac{3\pi}{8} + i\sin\frac{3\pi}{8}\right]^6 = \cos\left(\frac{18\pi}{8}\right) + i\sin\left(\frac{18\pi}{8}\right) \] \[ = \cos\frac{9\pi}{4} + i\sin\frac{9\pi}{4} \]

Step 3: Simplify angle. \[ \frac{9\pi}{4} = 2\pi + \frac{\pi}{4} \Rightarrow = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} \] \[ = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \]

Step 4: Multiply by \( \sqrt{2} \). \[ \sqrt{2} \cdot \left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}\right) = 1 + i \]