In Exercises 5-12, plot the point given in polar coordinates and find three additional polar represen- tations of the point, using $$ -2 \pi

Question

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

UpStudy AI · Accepted Answer

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
Here are the additional polar representations for each point within the range $$ -2\pi < \theta < 2\pi $$: 1. **$$ (3, \frac{5\pi}{6}) $$** - $$ (3, -\frac{7\pi}{6}) $$ - $$ (3, -\frac{\pi}{6}) $$ 2. **$$ (2, \frac{3\pi}{4}) $$** - $$ (2, -\frac{5\pi}{4}) $$ - $$ (2, -\frac{\pi}{4}) $$ 3. **$$ (-1, -\frac{\pi}{3}) $$** - $$ (-1, \frac{5\pi}{3}) $$ - $$ (-1, \frac{2\pi}{3}) $$ 4. **$$ (-3, -\frac{7\pi}{6}) $$** - $$ (-3, \frac{5\pi}{6}) $$ - $$ (-3, -\frac{\pi}{6}) $$ 5. **$$ (\sqrt{3}, \frac{5\pi}{6}) $$** - $$ (\sqrt{3}, -\frac{7\pi}{6}) $$ - $$ (\sqrt{3}, -\frac{\pi}{6}) $$ 6. **$$ (5\sqrt{2}, -\frac{11\pi}{6}) $$** - $$ (5\sqrt{2}, \frac{5\pi}{6}) $$ - $$ (5\sqrt{2}, -\frac{\pi}{6}) $$ 7. **$$ (\frac{3}{2}, -\frac{3\pi}{2}) $$** - $$ (\frac{3}{2}, \frac{\pi}{2}) $$ - $$ (\frac{3}{2}, \frac{5\pi}{2}) $$ These representations ensure that each point is expressed within the specified angle range.

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