Test Questions on Introduction to Polar Coordinates (Pre-Calculus)

📚 Quick Study Guide

• 🧭 Polar Coordinates: Represent a point in a plane using a distance $r$ from the origin (pole) and an angle $\theta$ from the positive x-axis (polar axis).
• 📍 Polar Coordinates Representation: $(r, \theta)$, where $r$ is the radial distance and $\theta$ is the angular coordinate.
• 🔄 Conversion Formulas:
  • ➡️ Cartesian to Polar: $r = \sqrt{x^2 + y^2}$, $\theta = \arctan(\frac{y}{x})$
  • ⬅️ Polar to Cartesian: $x = r \cos(\theta)$, $y = r \sin(\theta)$
• 📐 Common Polar Curves:
  • Circles: $r = a$ (centered at the pole), $r = 2a\cos(\theta)$ or $r = 2a\sin(\theta)$ (passing through the pole)
  • Lines: $\theta = a$ (passing through the pole), $r = a\sec(\theta)$ or $r = a\csc(\theta)$
  • Cardioids: $r = a(1 \pm \cos(\theta))$ or $r = a(1 \pm \sin(\theta))$
  • Lemniscates: $r^2 = a^2\cos(2\theta)$ or $r^2 = a^2\sin(2\theta)$
  • Roses: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$
• ➕ Symmetry Tests:
  • $\theta = \frac{\pi}{2}$ axis: Replace $(r, \theta)$ with $(r, \pi - \theta)$ or $(-r, -\theta)$.
  • Polar axis: Replace $(r, \theta)$ with $(r, -\theta)$ or $(-r, \pi - \theta)$.
  • Pole: Replace $(r, \theta)$ with $(r, \theta + \pi)$ or $(-r, \theta)$.

Practice Quiz

1. Question 1: What are the Cartesian coordinates of the point with polar coordinates $(2, \frac{\pi}{2})$?
   1. (A) (0, 2)
   2. (B) (2, 0)
   3. (C) (\sqrt{2}, \sqrt{2})
   4. (D) (-2, 0)
2. Question 2: Convert the Cartesian equation $x^2 + y^2 = 9$ to a polar equation.
   1. (A) $r = 3$
   2. (B) $r = 9$
   3. (C) $r^2 = 3$
   4. (D) $\theta = 3$
3. Question 3: Convert the polar equation $r = 4\cos(\theta)$ to a Cartesian equation.
   1. (A) $x^2 + y^2 = 4x$
   2. (B) $x^2 + y^2 = 4y$
   3. (C) $x^2 + y^2 = 16$
   4. (D) $x + y = 4$
4. Question 4: Which of the following points is equivalent to $(2, \frac{\pi}{3})$ in polar coordinates?
   1. (A) $(-2, \frac{4\pi}{3})$
   2. (B) $(-2, -\frac{\pi}{3})$
   3. (C) $(2, -\frac{\pi}{3})$
   4. (D) $(-2, \frac{2\pi}{3})$
5. Question 5: Identify the type of curve represented by the polar equation $r = 2 + 2\cos(\theta)$.
   1. (A) Circle
   2. (B) Line
   3. (C) Cardioid
   4. (D) Lemniscate
6. Question 6: Test the polar equation $r = 2\sin(\theta)$ for symmetry with respect to the polar axis.
   1. (A) Symmetric
   2. (B) Not Symmetric
   3. (C) Symmetry cannot be determined
   4. (D) Symmetric only for $r > 0$
7. Question 7: What is the value of $r$ at $\theta = \frac{\pi}{4}$ for the polar equation $r = 4\sin(2\theta)$?
   1. (A) 2
   2. (B) $2\sqrt{2}$
   3. (C) 4
   4. (D) $4\sqrt{2}$