| \(\theta \) | \({16\sin \theta ( \cos \theta +1)} \) | \(A(\theta )\) | |
|---|---|---|---|
| \(0\) | \(16\sin 0\left( \cos 0+1\right) =0\) | \(0\) | |
| \(\dfrac{\pi }{3}\) | \[ \begin{eqnarray*} 16&\sin \dfrac{\pi }{3}\left( \cos \dfrac{\pi }{3} +1\right) \\ &= 16\left(\dfrac{\sqrt{3}}{2}\right) \left(\dfrac{1}{2}+1\right) \\ &= 16\left( \dfrac{3\sqrt{3}}{4}\right) =12\sqrt{3} \end{eqnarray*} \] | \(\approx\!20.8\) | \(\longleftarrow\) absolute maximum value |
| \(\dfrac{\pi }{2}\) | \[ \begin{eqnarray*} 16&\sin \dfrac{\pi }{2}\left(\cos \dfrac{\pi }{2}+1\right) \\ &=16( 1) (0+1) =16 \end{eqnarray*} \] | \(16\) |