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The image contains a problem where rectangular coordinates are being converted to polar coordinates, with the results shown in both forms. Let's break down how polar coordinates \((r, \theta)\) are derived from rectangular coordinates \((x, y)\):

- **Formulas for Polar Conversion:**
  1. \( r = \sqrt{x^2 + y^2} \)
  2. \( \theta = \text{atan2}(y, x) \), where \( \theta \) is the angle (in radians) and is measured counterclockwise from the positive \( x \)-axis. The range for \( \theta \) is from 0 to \( 2\pi \).

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### Checking the Results from the Image:

1. **(a)** Rectangular: \( (0, 5) \)
   - \( r = \sqrt{0^2 + 5^2} = 5 \)
   - \( \theta = \text{atan2}(5, 0) = \frac{\pi}{2} \)
   - **Polar coordinates**: \( r = 5, \theta = \frac{\pi}{2} \)

2. **(b)** Rectangular: \( (-1, -\sqrt{3}) \)
   - \( r = \sqrt{(-1)^2 + (-\sqrt{3})^2} = 2 \)
   - \( \theta = \text{atan2}(-\sqrt{3}, -1) = \frac{4\pi}{3} \)
   - **Polar coordinates**: \( r = 2, \theta = \frac{4\pi}{3} \)

3. **(c)** Rectangular: \( (-\sqrt{3}, -1) \)
   - \( r = \sqrt{(-\sqrt{3})^2 + (-1)^2} = 2 \)
   - \( \theta = \text{atan2}(-1, -\sqrt{3}) = \frac{7\pi}{6} \)
   - **Polar coordinates**: \( r = 2, \theta = \frac{7\pi}{6} \)

4. **(d)** Rectangular: \( (4, -3) \)
   - \( r = \sqrt{4^2 + (-3)^2} = 5 \)
   - \( \theta = \text{atan2}(-3, 4) = -0.6435 \)
   - Since the angle needs to be in the range \( 0 \leq \theta < 2\pi \), add \( 2\pi \):
   - \( \theta = -0.6435 + 2\pi = 5.6397 \)
   - **Polar coordinates**: \( r = 5, \theta = 5.6397 \)

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Would you like more details or have any specific questions about this process?

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### Related Questions:
1. What are the key differences between polar and rectangular coordinate systems?
2. How is the angle \( \theta \) calculated using the arctangent function?
3. Why is the angle \( \theta \) adjusted to the range \( 0 \leq \theta < 2\pi \)?
4. Can you describe how polar coordinates are used in physics or engineering applications?
5. How would you convert from polar coordinates back to rectangular?

**Tip:** Always be mindful of the quadrant when converting from rectangular to polar coordinates, as it affects the angle.

Convert the following rectangular coordinates into polar coordinates. Always choose 0 ≤ θ < 2π.

Learn how to convert rectangular coordinates into polar coordinates with step-by-step solutions. Understand the formulas and theorems used, such as r = sqrt(x^2 + y^2) and θ = atan2(y, x). Suitable for students in Grades 10-12.

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