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To convert the polar equation to parametric equations, we can use the following relationships between Cartesian and polar coordinates: - \(x = r\cos\theta\) - \(y = r\sin\theta\) Given the equation \(r = 2 \sin (5\theta)\), we can substitute r into the parametric equations to get the following relationships: - \(x = (2 \sin (5\theta))\cos\theta\) - \(y = (2 \sin (5\theta))\sin\theta\)

To start creating a table of points, we will choose different angles for \(\theta\) (e.g., 0, \(\frac{\pi}{10}\), \(\frac{\pi}{5}\), \(\frac{3\pi}{10}\), \(\frac{2\pi}{5}\), \(\frac{\pi}{2}\), and \(\pi\)). Calculate the corresponding r values using the polar equation \(r = 2 \sin (5\theta)\), and then find the coordinate points (x, y) using the parametric equations we derived. | \(\boldsymbol{\theta}\) | \(\boldsymbol{r}\) | \(\boldsymbol{x}\) | \(\boldsymbol{y}\) | |-------------- |----- |----- |----- | | 0 | 0 | 0 | 0 | | \(\frac{\pi}{10}\) | ... | ... | ... | | \(\frac{\pi}{5}\) | ... | ... | ... | | \(\frac{3\pi}{10}\) | ... | ... | ... | | \(\frac{2\pi}{5}\) | ... | ... | ... | | \(\frac{\pi}{2}\) | 0 | 0 | 0 | | \(\pi\) | 0 | 0 | 0 | (This table will depend on the specific angles chosen and their respective calculated points.)

Using the table of points created in Step 2, plot each point on a polar grid by locating the angle \(\theta\) and the distance r along the radial direction from the origin.

To check the graph, we can use an online graphing calculator, a smartphone app, or a graphing calculator (such as Desmos, Wolfram, or a TI-83). Enter the polar equation \(r = 2 \sin (5\theta)\) and observe the resulting graph. It should match the plotted points from the table in Step 2.

After confirming that the points are correct, connect them smoothly to form the final graph of the polar equation \(r = 2 \sin (5\theta)\).