Certainly! Let's analyze and make a table for the polar equation: \[ r = 3 + 5\sin\theta \] ### Step 1: **Find where the graph crosses \( r = 0 \)** Set \( r = 0 \): \[ 0 = 3 + 5\sin\theta \implies \sin\theta = -\frac{3}{5} \] \[ \theta = \arcsin\left(-\frac{3}{5}\right) \approx -0.644 \text{ radians} \\ \] **Or in degrees:** \(\theta \approx -36.87^\circ\) Remember, since \(\sin\theta\) is negative, the other solution in \([0, 2\pi]\) is: \[ \theta = \pi - \arcsin\left(-\frac{3}{5}\right) = \pi + 0.644 \approx 3.785 \text{ radians} \\ \] **Or in degrees:** \(\theta \approx 216.87^\circ\) --- ### Step 2: **Make a Table of Values** Let's pick key values for \(\theta\): | \(\theta\) (degrees) | \(\theta\) (radians) | \(\sin\theta\) | \(r = 3 + 5\sin\theta\) | |:--------------------:|:-------------------:|:--------------:|:-----------------------:| | 0° | 0 | 0 | 3 | | 90° | \(\frac{\pi}{2}\) | 1 | 8 | | 180° | \(\pi\) | 0 | 3 | | 270° | \(\frac{3\pi}{2}\) | -1 | -2 | | 360° | \(2\pi\) | 0 | 3 | | -36.87° | \(-0.644\) | -0.6 | 0 | | 216.87° | \(3.785\) | -0.6 | 0 | --- ### Step 3: **Table Display** Here is the table in a clean format: \[ \begin{array}{|c|c|c|c|} \hline \theta\ (\text{deg}) & \theta\ (\text{rad}) & \sin\theta & r = 3 + 5\sin\theta \\ \hline 0^\circ & 0 & 0 & 3 \\ 90^\circ & \frac{\pi}{2} & 1 & 8 \\ 180^\circ & \pi & 0 & 3 \\ 270^\circ & \frac{3\pi}{2} & -1 & -2 \\ 360^\circ & 2\pi & 0 & 3 \\ -36.87^\circ & -0.644 & -0.6 & 0 \\ 216.87^\circ & 3.785 & -0.6 & 0 \\ \hline \end{array} \] --- ### Step 4: **Interpreting the Graph** - The maximum value of \( r \) is \( 8 \) (at \( 90^\circ \)). - The minimum value of \( r \) is \( -2 \) (at \( 270^\circ \)), which means the curve passes through the pole and continues out in the opposite direction. - The curve crosses the origin (pole) at \(\theta \approx -36.87^\circ\) and \(\theta \approx 216.87^\circ\). --- **This table will help you plot the graph accurately and see where the curve crosses \( r = 0 \).**