Trigonometry - Basics of Trigonometric Ratios

The following diagram shows a triagle \(\mathrm{ABC}\), where \(\mathrm{ABC}\), where \(\angle \mathrm{ACB}\) is \(90^{\circ}\), \(\mathrm{AB} = 3, \mathrm{AC} = 2\) and \(\angle \mathrm{BAC}\) is \(\theta\)

1. Show that \(\sin{\theta} = \frac{\sqrt{5}}{3}\).
2. Show that \(\sin {2\theta} = \frac{4\sqrt{5}}{9}\).
3. Find the exact value of \(\cos {2\theta}\).