Question

Real part of $\frac{1+\sin\frac{2\pi}{27}-i\cos\frac{2\pi}{27}}{1+\sin\frac{2\pi}{27}+i\cos\frac{2\pi}{27}}$ is equal to:

• A. $\cos\frac{2\pi}{27}$
• B. $\sin\frac{2\pi}{27}$
• C. $1+\sin\frac{2\pi}{27}$
• D. $1+\cos\frac{2\pi}{27}$
• E. $\sin\frac{2\pi}{27}+\cos\frac{2\pi}{27}$

Hint:

For complex numbers $w$, the real part is denoted as $Re(w)$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the property that $\frac{1+\sin\theta - i\cos\theta}{1+\sin\theta + i\cos\theta} = \sin\theta - i\cos\theta$.

Step 2: Analysis
Let $\theta = \frac{2\pi}{27}$. The expression simplifies to $\sin\theta - i\cos\theta$.

Step 3: Conclusion
The real part of this result is $\sin\theta = \sin\frac{2\pi}{27}$. Final Answer: (B)