Topic: Engineering Mathematics
The modulus of $1+\cos \alpha+i \sin \alpha$ is
- A. $2 \sin\frac{\alpha}{2}$
- B. $2 \cos\frac{\alpha}{2}$
- C. $\sin^{2}\frac{\alpha}{2}-1$
- D. $\cos^{2}\frac{\alpha}{2}-1$
Answer: B. $2 \cos\frac{\alpha}{2}$
Explanation: $z=1+\cos \alpha+i \sin \alpha$\n$|z|=\sqrt{(1+\cos \alpha)^{2}+\sin^{2}\alpha}$\n$=\sqrt{1+2 \cos \alpha+\cos^{2}\alpha+\sin^{2}\alpha}$\n$=\sqrt{2(1+\cos \alpha)}$\n$=\sqrt{2\times2 \cos^{2}(\frac{\alpha}{2})}=2 \cos(\frac{\alpha}{2})$