Trigonometry - Trigonometric Equations

Let \(f(x) = 6 + 6\sin{x}\). Part of the graph of \(f\) is shown below.

The shaded region is enclosed by the curve of \(f\), the \(x\)-axis, and the \(y\)-axis.

1. Solve for \(0 \leq x < 2\pi\).[5]
   1. \(6 + 6\sin{x} = 6\);
   2. \(6 + 6\sin{x} = 0\).
2. Write down the exact value of the \(x\)-intercept of \(f\), for \(0 \leq x < 2\).[1]
3. The area of the shaded region is \(k\). Find the value of \(k\), giving your answer in terms of \(\pi\).[6]

Let \(g(x) = 6 + 6\sin{\left(x - \frac{\pi}{2} \right)}\). The fraph of \(f\) is transformed to the graph of \(g\).

4. Give a full geometric description of this transformation.[2]
5. Given that \( \displaystyle \int_{p}^{p + \frac{3\pi}{2}} {g(x)} \mathrm{d}x = k \) and \( 0 \leq p < 2\pi\), write down the two values of \(p\).[3]