A) \[\frac{11\pi }{8},\,2\cos \left( \frac{\pi }{18} \right)\]
B) \[-\frac{7\pi }{18},-2\cos \left( \frac{11\pi }{18} \right)\]
C) \[\frac{2\pi }{9},2\cos \left( \frac{7\pi }{18} \right)\]
D) \[-\frac{\pi }{9},-2\cos \left( \frac{\pi }{18} \right)\]
Correct Answer: B
Solution :
\[\operatorname{z}=1+cos\,\frac{11\pi }{9}+i\,\sin \frac{11\pi }{9}\] \[\operatorname{Re}(z) > 0 and Im\left( z \right) < 0\], so the number lies in the fourth quadrant. Also \[z=2\cos \frac{11\pi }{18}\left\{ \cos \frac{11\pi }{18}+i\,\sin \frac{11\pi }{18} \right\}\] \[=\,\,2\cos \frac{11\pi }{18}\left\{ \cos \left( -\frac{7\,\pi }{18} \right)+i\,\sin \,\left( -\frac{7\,\pi }{18} \right) \right\}\] \[\therefore \,\,\,\arg (z)=-\frac{7\,\pi }{18}\] \[\left| z \right|=\left| 2\cos \frac{11\pi }{18} \right|=-2\cos \frac{11\pi }{18}\]
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