Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. $$r=2 \sin 5 \theta$$

Short answer

Question: Describe the process of graphing the polar equation \(r = 2 \sin (5\theta)\). Answer: To graph the polar equation \(r = 2 \sin (5\theta)\), follow these steps: 1. Convert the polar equation to parametric equations in terms of x and y. 2. Create a table of points using different angles (θ) and calculate the corresponding values of r, x, and y. 3. Plot the points on the polar grid. 4. Check the graph using a graphing utility. 5. Plot the final graph by connecting the points in a smooth curve.

Step-by-step solution

Convert the polar equation to parametric equations

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: - \(x = (2 \sin (5\theta))\cos\theta\) - \(y = (2 \sin (5\theta))\sin\theta\)

Create a table of points

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.)

Plot the points on the polar grid

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.

Check the graph using a graphing utility

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.

Plot the final graph

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