Test Questions on Converting Between Polar and Rectangular Coordinates

📚 Quick Study Guide

• 🧭 Polar Coordinates: Represent a point using a distance ($r$) from the origin and an angle ($\theta$) from the positive x-axis: $(r, \theta)$.
• 📏 Rectangular Coordinates: Represent a point using horizontal ($x$) and vertical ($y$) distances from the origin: $(x, y)$.
• 🔄 Conversion Formulas:
  • ➡️ Rectangular to Polar:
    • $r = \sqrt{x^2 + y^2}$
    • $\theta = \arctan(\frac{y}{x})$ (Adjust quadrant as needed!)
  • ⬅️ Polar to Rectangular:
    • $x = r \cos(\theta)$
    • $y = r \sin(\theta)$
• 💡 Key Considerations:
  • Always pay attention to the quadrant when finding $\theta$ using $\arctan$.
  • Remember that multiple polar coordinates can represent the same rectangular point (e.g., adding $2\pi$ to $\theta$).

✍️ Practice Quiz

1. Question 1: Convert the polar coordinates $(2, \frac{\pi}{2})$ to rectangular coordinates.
   1. (A) $(0, 2)$
   2. (B) $(2, 0)$
   3. (C) $(\sqrt{2}, \sqrt{2})$
   4. (D) $(-2, 0)$
2. Question 2: Convert the rectangular coordinates $(-1, 1)$ to polar coordinates.
   1. (A) $(\sqrt{2}, \frac{\pi}{4})$
   2. (B) $(\sqrt{2}, \frac{3\pi}{4})$
   3. (C) $(2, \frac{5\pi}{4})$
   4. (D) $(2, \frac{7\pi}{4})$
3. Question 3: Convert the polar coordinates $(4, \pi)$ to rectangular coordinates.
   1. (A) $(0, 4)$
   2. (B) $(4, 0)$
   3. (C) $(-4, 0)$
   4. (D) $(0, -4)$
4. Question 4: Convert the rectangular coordinates $(0, -3)$ to polar coordinates.
   1. (A) $(3, 0)$
   2. (B) $(3, \frac{\pi}{2})$
   3. (C) $(3, \frac{3\pi}{2})$
   4. (D) $(-3, \pi)$
5. Question 5: Convert the polar coordinates $(2\sqrt{2}, \frac{5\pi}{4})$ to rectangular coordinates.
   1. (A) $(2, 2)$
   2. (B) $(-2, -2)$
   3. (C) $(2, -2)$
   4. (D) $(-2, 2)$
6. Question 6: Convert the rectangular coordinates $(\sqrt{3}, -1)$ to polar coordinates.
   1. (A) $(2, \frac{\pi}{6})$
   2. (B) $(2, \frac{5\pi}{6})$
   3. (C) $(2, \frac{11\pi}{6})$
   4. (D) $(2, \frac{7\pi}{6})$
7. Question 7: Which of the following polar coordinate pairs represents the same point as the rectangular coordinates $(1, 0)$?
   1. (A) $(1, \frac{\pi}{2})$
   2. (B) $(1, \pi)$
   3. (C) $(1, 0)$
   4. (D) $(0, 1)$