Trigonometry

The diagram below shows a quadrilateral \(ABCD\) with obtuse angles \(A\hat{B}C\) and \(A\hat{D}C\).

\(AB = 5\) cm, \(BC = 4\) cm, \(CD = 4\) cm, \(AD = 4\) cm, \(B\hat{A}C = 30^{\circ}\), \(A\hat{B}C = x^{\circ}\), \(A\hat{D}C = y^{\circ}\).

1. Use the cosine rule to show that \(AC = \sqrt{41 - 40\cos{x}}\).[1]
2. Use the sine rule in triangle \(ABC\) to find another expression for \(AC\).[2]
3. 1. Hence, find \(x\), giving your answer to two decimal places.[6]
   2. Find \(AC\).
4. 1. Find \(y\).[5]
   2. Hence, or otherwise, find the area of triangle \(ACD\).