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Plot the points on the first circle, and find the 3 sets of polar coordinates.

Let's go through the solution based on the given polar coordinates \((3, \frac{5\pi}{4})\) and the instructions.

### 1. **Plotting the Point**

The point \((3, \frac{5\pi}{4})\) in polar coordinates is given by:
- \( r = 3 \): This is the radial distance from the origin.
- \( \theta = \frac{5\pi}{4} \): This is the angle measured counterclockwise from the positive \( x \)-axis.

To plot this point:
- Start at the origin and move 3 units outward.
- Locate the angle \( \frac{5\pi}{4} \), which is in the third quadrant (it’s slightly past \(\pi\), or 180°).
- Place the point 3 units out in this direction.

### 2. **Finding Additional Polar Coordinates**

#### Condition (a): \( r > 0 \) and \( -2\pi \leq \theta < 0 \)
For a positive radius (\(r = 3\)) and a negative angle, we can find an equivalent angle by subtracting \(2\pi\) from \( \frac{5\pi}{4} \):
\[
\theta = \frac{5\pi}{4} - 2\pi = \frac{5\pi}{4} - \frac{8\pi}{4} = -\frac{3\pi}{4}
\]
Thus, one equivalent set of coordinates is:
\[
\left( 3, -\frac{3\pi}{4} \right)
\]

#### Condition (b): \( r < 0 \) and \( 0 \leq \theta < 2\pi \)
For a negative radius, we need to reverse the direction. So, \( r = -3 \). To keep the point in the same location, add \(\pi\) to the angle \( \frac{5\pi}{4} \):
\[
\theta = \frac{5\pi}{4} + \pi = \frac{5\pi}{4} + \frac{4\pi}{4} = \frac{9\pi}{4}
\]
Since \(\frac{9\pi}{4}\) is greater than \(2\pi\), subtract \(2\pi\) to bring it back within the range \([0, 2\pi)\):
\[
\frac{9\pi}{4} - 2\pi = \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}
\]
Thus, the second set of coordinates is:
\[
\left( -3, \frac{\pi}{4} \right)
\]

### Summary of Solutions:
1. Original point: \((3, \frac{5\pi}{4})\)
2. For condition (a): \((3, -\frac{3\pi}{4})\)
3. For condition (b): \((-3, \frac{\pi}{4})\)

Would you like further clarification on these steps?

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Here are five related questions to deepen your understanding:
1. How can we convert any polar coordinate with a positive radius to one with a negative radius?
2. What are the steps for converting angles in polar coordinates between positive and negative values?
3. How can you determine the quadrant of a point using polar coordinates?
4. Why is subtracting \(2\pi\) effective in finding equivalent angles?
5. What is the significance of using \( \theta = \theta + \pi \) when switching the sign of \( r \)?

**Tip:** In polar coordinates, adding or subtracting \(2\pi\) to an angle gives an equivalent point with the same position, due to the circular nature of polar coordinates.

Plot the point whose polar coordinates are (3, 5π/4). Find two sets of polar coordinates for the point (3, 5π/4) with the criteria: a. r > 0 and -2π ≤ θ < 0; b. r < 0 and 0 ≤ θ < 2π.

Explore how to plot the polar coordinates (3, 5π/4) and find equivalent coordinates under specified conditions. This problem involves understanding polar coordinates, angle conversions, and theorems related to equivalent angles. Suitable for high school students in Grades 10-12.

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