2023 May TZ2 P1 Q8

Consider an acute angle \(\theta\) such that \(\cos{\theta} = \frac{2}{3}\).

1. Find the value of

   1. \(\sin{\theta}\);

   2. \(\sin{2\theta}\).

      [4]

The following diagram shows triangle \(\mathrm{ABC}\), with \(\hat{B} = \theta\), \(\hat{A} = 2\theta\), \(\mathrm{BC} = a\) and \(\mathrm{AC} = b\).

2. show that \(b = \frac{3a}{4}\).

   [2]

\([\mathrm{BA}]\) is extended to form an isosceles triangle \(\mathrm{DAC}\), with \(\hat{D} = \theta\), as shown in the following diagram.

3. Find the value of \(\sin{\mathrm{C\hat{A}D}}\).

   [3]

4. Find the area of triangle \(\mathrm{DAC}\), in terms of \(a\).

   [5]