Answer
Graph estimate: $0.77,$
Exact value: $\displaystyle \frac{4\sqrt{3}}{9}$
Work Step by Step
Estimating from the graph, the highest points have $y \approx 0.77$.
For the exact value, we solve $\displaystyle \frac{dy}{d\theta}=0$.
$y =r\sin\theta=\sin\theta\sin 2\theta \qquad(*)$
$\displaystyle \frac{dy}{d\theta} =2\sin\theta\cos 2\theta+\cos\theta\sin 2\theta$
... use the double angle identities ..
$=2\sin\theta(2\cos^{2}\theta-1)+\cos\theta(2\sin\theta\cos\theta)$
$=2\sin\theta(2\cos^{2}\theta-1)+2\sin\theta\cos^{2}\theta$
$=2\sin\theta(3\cos^{2}\theta-1)$
In the first quadrant, $3\cos^{2}\theta-1$= $0$
when $\displaystyle \cos\theta=\frac{1}{\sqrt{3}} \qquad $
$\sin\theta=\sqrt{1-\cos^{2}\theta}=\sqrt{1-\frac{1}{3}}=\sqrt{\frac{2}{3}} \Leftrightarrow$
We go back to (*)
$ y=\sin\theta\sin 2\theta$
$=\sin\theta$($2\sin\theta\cos\theta)$
$=2\sin^{2}\theta\cos\theta$
$=2\displaystyle \cdot\frac{2}{3}\cdot\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{4\sqrt{3}}{9}$
$(\approx 0.77)$.
