5. In this question you must show detailed reasoning.
- Given that
$$z _ { 1 } = 6 \left( \cos \left( \frac { \pi } { 3 } \right) + i \sin \left( \frac { \pi } { 3 } \right) \right) \quad \text { and } \quad z _ { 2 } = 6 \sqrt { 3 } \left( \cos \left( \frac { 5 \pi } { 6 } \right) + i \sin \left( \frac { 5 \pi } { 6 } \right) \right)$$
show that
$$z _ { 1 } + z _ { 2 } = 12 \left( \cos \left( \frac { 2 \pi } { 3 } \right) + i \sin \left( \frac { 2 \pi } { 3 } \right) \right)$$
- Given that
$$\arg ( z - 5 ) = \frac { 2 \pi } { 3 }$$
determine the least value of \(| \boldsymbol { z } |\) as \(Z\) varies.
[0pt]
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