Pre-Calculus: Step-by-Step Examples for Plotting Polar Points

📚 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).
📐
• Coordinate Pair: A polar point is given as $(r, \theta)$, where $r$ is the radial distance and $\theta$ is the angle in radians or degrees.
➕
• Positive $r$: If $r > 0$, the point lies on the terminal side of the angle $\theta$.
➖
• Negative $r$: If $r < 0$, the point lies on the ray opposite the terminal side of the angle $\theta$. This is equivalent to adding or subtracting $\pi$ (or $180^{\circ}$) to the angle.
🔄
• Multiple Representations: Each point in the polar coordinate system has infinitely many representations, since adding multiples of $2\pi$ (or $360^{\circ}$) to $\theta$ results in the same point. Also, $(-r, \theta)$ is the same point as $(r, \theta + \pi)$.
📍
• Plotting Steps: To plot $(r, \theta)$, first find the angle $\theta$, then move along that line a distance of $|r|$. If $r$ is negative, move in the opposite direction.

📝 Practice Quiz

1. What quadrant does the point $(2, \frac{\pi}{4})$ lie in?
   1. I
   2. II
   3. III
   4. IV
2. Where is the point $(-3, \pi)$ located?
   1. On the positive x-axis
   2. On the negative x-axis
   3. On the positive y-axis
   4. On the negative y-axis
3. Which point is equivalent to $(4, \frac{\pi}{2})$?
   1. $(-4, \frac{3\pi}{2})$
   2. $(-4, -\frac{\pi}{2})$
   3. $(4, -\frac{\pi}{2})$
   4. $(4, \frac{5\pi}{2})$
4. What is the location of the point $(0, \frac{\pi}{3})$?
   1. Origin
   2. On the positive x-axis
   3. On the positive y-axis
   4. Undefined
5. Which of the following points is equivalent to $(-2, \frac{\pi}{6})$?
   1. $(2, \frac{7\pi}{6})$
   2. $(2, \frac{\pi}{6})$
   3. $(-2, \frac{7\pi}{6})$
   4. $(2, -\frac{\pi}{6})$
6. In which quadrant does the point $(-1, \frac{5\pi}{4})$ lie?
   1. I
   2. II
   3. III
   4. IV
7. What is the distance from the origin to the point $(5, \frac{3\pi}{2})$?
   1. 3
   2. 5
   3. 0
   4. -5