Solved examples: Multiple representations of a polar point

📚 Quick Study Guide

• 🧭 Polar Coordinates: A point $P$ in the polar coordinate system is represented by $(r, \theta)$, where $r$ is the directed distance from the pole (origin) to $P$, and $\theta$ is the directed angle, measured counterclockwise, from the polar axis to the line segment $OP$.
• 🔄 Multiple Representations: Unlike Cartesian coordinates, a polar point has infinitely many representations. Adding or subtracting multiples of $2\pi$ to $\theta$ does not change the point. Also, $(r, \theta)$ is the same as $(-r, \theta + \pi)$.
• ➕ Adding $2\pi$: $(r, \theta) = (r, \theta + 2\pi n)$ for any integer $n$.
• ➖ Negative $r$: $(r, \theta) = (-r, \theta + (2n+1)\pi)$ for any integer $n$.
• 📐 Converting to Cartesian: $x = r \cos(\theta)$ and $y = r \sin(\theta)$. This is useful for checking if different polar coordinates represent the same point.
• 💡 Tips for Finding Representations: Given a point $(r, \theta)$, try adding $2\pi$ or subtracting $2\pi$ from $\theta$. Also, consider using $-r$ and adjusting $\theta$ accordingly.

Practice Quiz

1. Which of the following polar coordinates represents the same point as $(2, \frac{\pi}{3})$?

   1. $(2, \frac{7\pi}{3})$
   2. $(-2, \frac{4\pi}{3})$
   3. $(2, -\frac{\pi}{3})$
   4. $(-2, -\frac{\pi}{3})$

2. Which of the following is NOT a valid polar representation of the point $(3, \frac{\pi}{2})$?

   1. $(3, \frac{5\pi}{2})$
   2. $(-3, -\frac{\pi}{2})$
   3. $(-3, \frac{3\pi}{2})$
   4. $(3, -\frac{3\pi}{2})$

3. The polar coordinates $(-4, \frac{\pi}{6})$ are equivalent to which of the following?

   1. $(4, \frac{7\pi}{6})$
   2. $(4, -\frac{\pi}{6})$
   3. $(-4, \frac{13\pi}{6})$
   4. $(4, \frac{11\pi}{6})$

4. Which coordinate is equivalent to $(r, \theta)$?

   1. $(-r, \theta)$
   2. $(-r, \theta - \pi)$
   3. $(-r, -\theta)$
   4. $(r, -\theta)$

5. Which of the following represents the same point as $(-1, 0)$?

   1. $(1, \pi)$
   2. $(-1, 2\pi)$
   3. $(1, 0)$
   4. $(-1, -\pi)$

6. Given the point $(\sqrt{2}, \frac{3\pi}{4})$, find an equivalent representation with a positive $r$ and a negative angle.

   1. $(\sqrt{2}, -\frac{5\pi}{4})$
   2. $(\sqrt{2}, -\frac{\pi}{4})$
   3. $(-\sqrt{2}, -\frac{\pi}{4})$
   4. $(-\sqrt{2}, \frac{\pi}{4})$

7. Which of the following is a representation of $(5, \frac{2\pi}{3})$ with a negative $r$ value?

   1. $(-5, \frac{2\pi}{3})$
   2. $(-5, -\frac{\pi}{3})$
   3. $(-5, \frac{5\pi}{3})$
   4. $(-5, \frac{-\pi}{3})$