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Q.

The principle value of the arg(z) and $| z |$ of the complex number $Z = 1 + \cos \frac{11 π}{9} + i \sin \frac{11 π}{9}$ are respectively

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a

$\frac{2 π}{9}, 2 \cos \frac{7 π}{18}$

b

$\frac{11 π}{18}, 2 \cos \frac{π}{18}$

c

$\frac{- 7 π}{18}, 2 \cos \frac{7 π}{18}$

d

$\frac{- π}{9}, - 2 \cos \frac{π}{8}$

answer is B.

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Detailed Solution

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$\begin{array}{l} Z = 2 \cos \frac{11 π}{18} [\cos \frac{11 π}{18} + i \sin \frac{11 π}{18}] or - 2 \cos \frac{7 π}{18} [- \cos \frac{7 π}{18} + i \sin \frac{7 π}{18}] \\ = 2 \cos \frac{7 π}{18} [\cos \frac{7 π}{18} - i \sin \frac{7 π}{18}] \end{array}$

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The principle value of the arg(z) and |z| of the complex number Z=1+cos11π9+isin11π9 are respectively