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 $2\pi$.

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: $r = 5, \theta = 5.6397$

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

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.