91Ó°ÊÓ

For polar graphs, we look for symmetry with respect to the polar axis, the line \(\theta=\pi\), or the pole. We check for symmetry by performing the following tests: - Replace \(\theta\) with \(-\theta\). If the given equation is unchanged, then our curve is symmetric about the polar axis. In our case, we have $$r=\sin(-3\theta)=\sin(3\theta)$$ so we have symmetry about the polar axis. - Replace \(\theta\) with \(\theta+\pi\). If the given equation is unchanged, then our curve is symmetric about the line \(\theta=\pi\). In our case, $$r=\sin[3(\theta+\pi)]=\sin(3\theta+3\pi)$$ which is different, so our curve is not symmetric about the line \(\theta=\pi\). - Replace \(r\) with \(-r\). If a valid point exists on the given equation, then our curve is symmetric about the pole. In our case, $$-r=\sin(3\theta) \Rightarrow r=-\sin(3\theta)$$ which is not the original equation so it's not symmetric about the pole.

Polar equations usually have a range of values for \(\theta\) that cover one period of the curve. Since our equation involves the sine function multiplied by \(3\theta\), the period of the curve is \(\frac{2\pi}{3}\). We can define the range of \(\theta\) as \([0, \frac{2\pi}{3}]\).

To plot the curve, we will plot points for key angles in the range \([0, \frac{2\pi}{3}]\). For each \(\theta\), calculate corresponding \(r\) value and plot the point in polar coordinates. Here are some key angles and corresponding \(r\) values: $$\begin{array}{|c|c|} \hline \theta& r\\ \hline 0 & \sin(3 \cdot 0) = 0\\ \hline \frac{\pi}{6} & \sin(3 \cdot \frac{\pi}{6}) = \sin(\frac{\pi}{2}) = 1\\ \hline \frac{\pi}{3} & \sin(3 \cdot \frac{\pi}{3}) = \sin(\pi) = 0\\ \hline \frac{\pi}{2} & \sin(3 \cdot \frac{\pi}{2}) = \sin(\frac{3\pi}{2}) = -1\\ \hline \frac{2\pi}{3} & \sin(3 \cdot \frac{2\pi}{3}) = \sin(2\pi) = 0\\ \hline \end{array}$$ Plot these points and sketch the curve that goes through them, remembering to incorporate the symmetry properties.

Input the polar equation \(r=\sin 3\theta\) into your preferred graphing utility (like a graphing calculator or software like Desmos). Make sure the range of theta is set to \([0, \frac{2\pi}{3}]\), and compare the graph with your hand-drawn sketch to verify its accuracy. If necessary, adjust your sketch to match the graph generated by the graphing utility. Finally, your finished graph should resemble a three-petaled flower, with petals oriented at \(120^\circ\) intervals and symmetric about the polar axis.