Trigonometry - Trigonometric Equations

Let \(f(x) = \sin^3{x}\).

1. 1. Write down the range of the function \(f\).
   2. Consider \(f(x) -1, 0 \leq x \leq 2\pi\). Wirte down the number of solutions to this equation.
      Justify your answer.
2. Find \(f'(x)\), giving your answer in the form \(a\sin^p{x} \cos^1{x}\) where \(a, p, q, \in \mathbb{Z}\).
3. Let \(g(x) = \sqrt{3} \sin{x} (\cos{x})^{\frac{1}{2}}\) for \(0 \leq x \leq \frac{\pi}{2}\). Find the volume generated when the curve of \(g\) is revolved through \(2\pi\) about the \(x\)-axis.