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Mathematics Question on complex numbers

The principal argument of the complex numb Z=1+sin⁡π3+icos⁡π31+sin⁡π3−icos⁡π3Z=\frac{1+\sin \frac{\pi}{3}+i \cos\frac{\pi}{3} }{1+\sin \frac{\pi}{3} - i \cos\frac{\pi}{3} } is

A

π3\frac{\pi}{ 3}

B

π6\frac{\pi}{6 }

C

2π3\frac{2\pi}{ 3}

D

π2\frac{\pi}{2 }

Answer

π6\frac{\pi}{6 }

Explanation

Solution

arg⁡(z)=arg⁡(\arg (z)=\arg ( Numerator )−arg⁡()-\arg ( Denominator ))
=tan⁡−1∣cos⁡π31+sin⁡π3∣+tan⁡−1∣cos⁡π31+sin⁡π3∣=\tan ^{-1}\left|\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right|+\tan ^{-1}\left|\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right|
=2tan⁡−1[cos⁡π31+sin⁡π3]=2 \tan ^{-1}\left[\frac{\cos \frac{\pi}{3}}{1+\sin \frac{\pi}{3}}\right]
=2tan⁡−1(2−3)=2 \tan ^{-1}(2-\sqrt{3})
=2×15∘=30∘=π6=2 \times 15^{\circ}=30^{\circ}=\frac{\pi}{6}