2 The acute angles \(\alpha\) and \(\beta\) are given by \(\tan \alpha = \frac { 2 } { \sqrt { 5 } }\) and \(\tan \beta = \frac { 1 } { 2 }\).
- Show that \(\sin \alpha = \frac { 2 } { 3 }\), and find the exact value of \(\cos \alpha\).
- Hence find the exact value of \(\sin 2 \alpha\).
- Show that the exact value of \(\cos ( \alpha - \beta )\) can be expressed as \(\frac { 2 } { 15 } ( k + \sqrt { 5 } )\), where \(k\) is an integer.