There is a unique complex number \(w\) that satisfies both
$$| w - 3 | = 2 \quad \text { and } \quad \arg ( w + 1 ) = \alpha$$
where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
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Find the value of \(\alpha\).
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(ii) Express \(w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
Give each of \(r\) and \(\theta\) to two significant figures.
(a) Show that
$$\frac { 1 } { r + 2 } - \frac { 1 } { r + 3 } = \frac { 1 } { ( r + 2 ) ( r + 3 ) }$$