It is given that \(- 1 + ( \sqrt { } 5 ) \mathrm { i }\) is a root of the equation \(z ^ { 3 } + 2 z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation.
The complex number \(w\) has modulus 1 and argument \(2 \theta\) radians. Show that \(\frac { w - 1 } { w + 1 } = \mathrm { i } \tan \theta\).