In Exercises 5-12, plot the point given in polar coordinates and find three additional polar represen- tations of the point, using \( -2 \pi<\theta<2 \pi \) 5. \( \left(3, \frac{5 \pi}{6}\right) \) \( \begin{array}{ll}\text { 7. }\left(-1,-\frac{\pi}{3}\right) & \text { 6. }\left(2, \frac{3 \pi}{4}\right) \\ \text { 9. }\left(\sqrt{3}, \frac{5 \pi}{6}\right) & \text { 8. }\left(-3,-\frac{7 \pi}{6}\right) \\ \text { 11. }\left(\frac{3}{2},-\frac{3 \pi}{2}\right) & \text { 10. }\left(5 \sqrt{2},-\frac{11 \pi}{6}\right)\end{array} \)

To solve the problem, we will first plot the given polar coordinates and then find three additional polar representations for each point within the range \( -2\pi < \theta < 2\pi \). ### Step 1: Plot the Point The polar coordinates are given in the form \( (r, \theta) \), where \( r \) is the radius and \( \theta \) is the angle. 1. **For \( (3, \frac{5\pi}{6}) \)**: - The radius \( r = 3 \) means we move 3 units from the origin. - The angle \( \theta = \frac{5\pi}{6} \) is in the second quadrant. ### Step 2: Find Additional Polar Representations To find additional representations of the point, we can use the following properties of polar coordinates: - Adding or subtracting \( 2\pi \) to the angle \( \theta \) gives the same point. - If \( r \) is negative, the angle can be adjusted by adding or subtracting \( \pi \). #### Finding Three Additional Representations for \( (3, \frac{5\pi}{6}) \): 1. **First Representation**: \[ (3, \frac{5\pi}{6} + 2\pi) = (3, \frac{5\pi}{6} + \frac{12\pi}{6}) = (3, \frac{17\pi}{6}) \] (This is outside the range, so we will not use it.) 2. **Second Representation**: \[ (3, \frac{5\pi}{6} - 2\pi) = (3, \frac{5\pi}{6} - \frac{12\pi}{6}) = (3, -\frac{7\pi}{6}) \] 3. **Third Representation**: \[ (3, \frac{5\pi}{6} - \pi) = (3, \frac{5\pi}{6} - \frac{6\pi}{6}) = (3, -\frac{\pi}{6}) \] 4. **Fourth Representation**: \[ (3, \frac{5\pi}{6} + \pi) = (3, \frac{5\pi}{6} + \frac{6\pi}{6}) = (3, \frac{11\pi}{6}) \] (This is also outside the range, so we will not use it.) ### Summary of Representations for \( (3, \frac{5\pi}{6}) \): - \( (3, \frac{5\pi}{6}) \) - \( (3, -\frac{7\pi}{6}) \) - \( (3, -\frac{\pi}{6}) \) ### Step 3: Repeat for Other Points Now, let's find the additional representations for the other points in the list. #### 6. \( (2, \frac{3\pi}{4}) \) 1. **First Representation**: \[ (2, \frac{3\pi}{4} + 2\pi) = (2, \frac{3\pi}{4} + \frac{8\pi}{4}) = (2, \frac{11\pi}{4}) \quad \text{(out of range)} \] 2. **Second Representation**: \[ (2, \frac{3\pi}{4} - 2\pi) = (2, \frac{3\pi}{4} - \frac{8\pi}{4}) = (2, -\frac{5\pi}{4}) \] 3. **Third Representation**: \[ (2, \frac{3\pi}{4} - \pi) = (2, \frac{3\pi}{4} - \frac{4\pi}{4}) = (2, -\frac{\pi}{4}) \] #### 7. \( (-1, -\frac{\pi}{3}) \) 1. **First Representation**: \[ (-1, -\frac{\pi}{3} + 2\pi) = (-1, -\frac{\pi}{3} + \frac{6\pi}{3}) = (-1, \frac{5\pi}{3}) \] 2. **Second Representation**: \[ (-1, -\frac{\pi}{3} - 2\pi) = (-1, -\frac{\pi}{3} - \frac{6\pi}{3}) = (-1, -\frac{7\pi}{3}) \quad \text{(out of range)} \] 3. **Third Representation**: \[ (-1, -\frac{\pi}{3} + \pi) = (-1, -\frac{\pi}{3} + \frac{3\pi}{3}) = (-1, \frac{2\pi}{3}) \] #### 8. \( (-3, -\frac{7\pi}{6}) \) 1. **First Representation**: \[ (-3, -\frac{7\pi}{6} + 2\pi) = (-3, -\frac{7\pi}{6} + \frac{12\pi}{6}) = (-3, \frac{5\pi}{6}) \] 2. **Second Representation**: \[ (-3, -\frac{7\pi}{6} - 2\pi) = (-3, -\frac{7\pi}{6} - \frac{12\pi}{6}) = (-3, -\frac{19\pi}{6}) \quad \text{(out of range)} \] 3. **Third Representation**: \[ (-3, -\frac{7\pi}{6} + \pi) = (-3, -\frac{7\pi}{6} + \frac{6\pi}{6}) = (-3, -\frac{\pi}{6}) \] #### 9. \( (\sqrt{3}, \frac{5\pi}{6}) \) 1. **First Representation**: \[ (\sqrt{3}, \frac{5\pi}{6} + 2\pi) = (\sqrt{3}, \frac{5\pi}{6} + \frac{12\pi}{6}) = (\sqrt{3}, \frac{17\pi}{6}) \quad \text{(out of range)} \] 2. **Second Representation**: \[ (\sqrt{3}, \frac{5\pi}{6} - 2\pi) = (\sqrt{3}, \frac{5\pi}{6} - \frac{12\pi}{6