8 Abdoallah wants to write the complex number \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) Here is his method: $$\begin{array} { r l r l } r & = \sqrt { ( - 1 ) ^ { 2 } + ( \sqrt { 3 } ) ^ { 2 } } & & \tan \theta = \frac { \sqrt { 3 } } { - 1 }
& = \sqrt { 1 + 3 } & & \Rightarrow
& = \sqrt { 4 } & & \tan \theta = - \sqrt { 3 }
& = 2 & & \theta = \tan ^ { - 1 } ( - \sqrt { 3 } )
& & \theta = - \frac { \pi } { 3 }
& - 1 + i \sqrt { 3 } = 2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right) \end{array}$$ There is an error in Abdoallah's method. 8

1. Show that Abdoallah's answer is wrong by writing $$2 \left( \cos \left( - \frac { \pi } { 3 } \right) + i \sin \left( - \frac { \pi } { 3 } \right) \right)$$ in the form \(x + \mathrm { i } y\)
   Simplify your answer.
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2. Explain the error in Abdoallah's method.
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3. Express \(- 1 + \mathrm { i } \sqrt { 3 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\)
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4. Write down the complex conjugate of \(- 1 + i \sqrt { 3 }\)